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NOTES ON LIFE INSURANCE. 



PART FIRST— THEORETICAL. 
PART SECOND— PRACTICAL. 

WITH APPENDIX. 



" The rate of premium which must be charged in order to carry out an insurance con- 
tract. i<? the problem which stands at the threshold of Life Assurance." 

Dr. Farr. 



SECOND EDITION 

EVISED, ENLARGED, AND RE-ARRANGED. 



BY 



V* GTTSTAVUS W. SMITH. 







OFCq> 

»ht¥ 






V 



NEW-YORK: 

S. W. GREEN, PRINTER, Nos. 16 & 18 JACOB STREET. 

1875. 



//4 w 



Entered, according to Act of Congress in the year 1875, by 

GUSTAVUS W. SMITH, 
in the Office of the Librarian of Congress, at Washington. 



" Does the system itself rest on principles and laws so certain and stable as 
to justify a reasonable conviction that if the system is fairly and honestly ad- 
ministered, the bread that is cast on its waters will be surely found, though 
after many days ?" (John E. Sanford, 1868.) 

"What is wanted is that the school-house and the press, the universal edu- 
cators, shall take up the matter, not in the interests of companies or their 
agents, but in that of the public and its coming generations. The companies 
have nothing to fear but every thing to hope from the most thorough discus- 
sion of their plans and the exposure of all the details of their management." 

(Elizur Wright, 1872.) 



PTJBLISHEES* NOTICES. 



The following extracts are from some of the many letters ad- 
dressed to the author of Notes on Life Insurance, soon after the 
publication of the first edition, 1870. 

Mr. John Paterson, mathematician, Insurance Department, State 
of Now- York, writes : 

" I hesitate not to say that, in matter and manner, the work opportunely 
meets a great public want. To the unmathematical portion of the business 
community, who may feel interested, or even only curious, about this great 
financiering speculation, which has so recently, as it were, flooded almost 
the entire continent, it reveals in plain words, by successive steps, the entire 
process of management, and dispels all the presupposed mystery in which 
the question was involved — all this by ordinary arithmetic, the well-selected 
examples in which are clearly solved. Further than this, the algebraic 
deductions will bear the strictest scrutiny, and might well serve to initiate 
the tyro in actuarial science, while the advanced student may find some 
corroboration in the author's critical views. It is a household word with 
him, that what is usually termed the reserve is a 'Trust Fund Deposit,' 
does not in any sense belong to the company, but is a debt due to the policy- 
holder. 

" The comments in the second part of the treatise deserve the thoughtful con- 
sideration of all officials, as well as incumbents, who stand under the banner of 
this mammoth institution of the age." 

Mr. H. A. Griswold, of Louisville, Ky., writes : 

" I have read your Notes on Life Insurance with the greatest interest, and 
congratulate you on the ability with which a subject, usually so intricate and 
incomprehensible, is made intelligible to ordinary minds. No great mathema- 
tical knowledge is required : the algebra is within the scope of ordinary school 
instruction, and even this moderate amount is not necessary, provided the in- 
quirer will assume the accuracy of a few formulae." 

Professor William H. C. Bartlett, Actuary of the Mutual Life 
Insurance Company of New- York, says : 

' ' It gives me great pleasure to say that the Notes on Life Insurance, by 
General G. W. Smith, is a very valuable contribution to our stock of informa- 
tion on the subject, and I very heartily commend the work to public favor. It 
should be in the hands of every insurance agent in the country. " 



G NOTES ON LIFE INSURANCE. 

General Robert Toombs. Georgia, says : 

" I have read your book with great pleasure, and consider it a very valuable 
addition to the science of Life Insurance, now greatly needed, especially by the 
people — I mean policy-holders — and I doubt not will be of incalculable benefi: 
to them. 

" You have given a simple rule, simplified, by which to enable thousands of 
policy-holders to know something of the condition of companies in which such 
vast amounts of their annual earnings are invested for the noblest of purposes, 
and with the means of arresting gradual decay and ruin of such of these 
companies as may be badly managed. I am very anxious to see it largely 
circulated." 

Hon. Oliver Pillsbubt, Insurance Commissioner of New-Hamp- 
shire, says : 

" I have been much interested in your ' Notes.' They are all that I expected 
and more. I wish they could be scattered as leaves of the forest, all abroad. 
You have literally turned the mysteries of Life Insurance inside out, and ren- 
dered them susceptible of comprehension by ordinary reflecting minds. I con- 
sider the unpretending pamphlet worth more to me than all the elaborated 
volumes I have ever purchased. Your style is peculiarly adapted to the com- 
mon people." 

Prof. A. L. Perky, of Williams College, says : 

"' I have read this work with great care, and I may add, with great interest. 
It has cleared up to my mind a subject of which I was almost totally ignorant 
before, although I have had three policies on my life for several years. In 
taking them out, and paying the annual premiums, I have walked by faith and 
not by sight. I believe I know now the whys and wherefores, as well as the 
modus operandi" 

The Hon. Lixtox Stephens, Georgia, says : 

"Your Notes on Life Insurance go to the very bottom of a knowledge 
which has heretofore been a sealed book to all but the initiated few, and places 
it within the easy comprehension of all intelligent men of business. Even per- 
sons of quite limited education can acquire from this most useful little work a 
sufficient acquaintance with the principles of insurance to enable them to judge 
for themselves of the trustworthiness of the multitude of different insurance 
institutions which are now claiming the confidence and struggling for the pa- 
tronage of the public. The importance of this timely work is to be measured 
only by the present vast and still-increasing magnitude of the business of Life 
Insurance." 

Prof. David Murray, of Rutgers College, New-Brunswick, 
1ST. J., says : 

"I have read the Notes on Life Insurance with the greatest interest. I am 
free to say that I consider it the best popular explanation of the theory and 
practice of Life Insurance that I know. I have examined, with constantly in- 
creasing admiration, the lucid development of, to most, a complicated and dif- 



NOTES ON LIFE INSURANCE. 7 

ficult subject. It seems to me exactly fitted to be put into the bauds of the of- 
ficers and agents of all our life companies, in order that they may be fur- 
nished with something more than a mere 'routine' knowledge of their busi- 



General A. P. Stewart, Assistant Actuary of the St. Louis Mu- 
tual Life Insurance Company, says : 

" I have read the ' Notes ' very carefully. The discussion of the mathemati- 
cal theory of Life Insurance is the simplest and clearest I have seen, and would 
be a good text-book on the subject for schools and colleges. The practical part 
of the ' Notes ' is also very valuable, and should be read by every one who is 
insured or who contemplates insuring." 

Hon. William Barnes, former Superintendent of the Insurance 
Department of the State of New- York, says : 

" In my opinion, it is the most useful and valuable work ever issued in this 
country for the purpose of popular information in the mathematics and funda- 
mental principles of Life Insurance. Indeed, I am not familiar with any for- 
eign book which attains the object so fully and completely." 

General S. B. Btjckner, Louisville, Ky., says : 

" I can not adequately commend your Notes on Life Insurance. Though 
not addressed to insurance people, I consider them, in the present condition of 
the business, almost indispensable to agents. Nowhere else in the English 
language can be found so complete and so intelligible an epitome of all the 
principles which constitute the basis of Life Insurance. In the hands of any 
intelligent reader, they will unlock the mysteries in which some seek to vail 
the business which should be exposed in the clearest light." 



CONTENTS. 



PART I. THEORY. (Arithmetical Discussion.) 

Chapter I. Net Premiums. 

Chapter II. Construction of Commutation Columns. 

Chapter III. Trust Fund Deposit or " Reserve." 

Chapter IV. Amount at Risk. Valuation of Policies. 

Chapter V. Joint Lives. 



Part I. Continued. (Algebraic Discussion.) 

Chapter VI. Net Premiums. 

Chapter VII. Trust Fund Deposit or " Reserve." 

Chapter VIII. Annuities Paid Oftener than Once a Year. 



PART II. PRACTICAL LIFE INSURANCE. 

Chapter IX. General Management. 
Chapter X. Variety in Plans of Insurance. 
Chapter XI. Gross Valuations. Net Valuations. 

Chapter XII. Disposition made of Deposit when Renewal Premium is not 
Paid. Annual Statements. 



APPENDIX. 

Chapter XIII. Extracts, 1783, and Quotations, 1870 and 1871. 
Chapter XIV. Algebraic Summary, Formulas, and Tables. 



PREFACE 



The numerical examples and tables in the text are simply illus- 
trative ; they are not intended as "sums to bo worked and the 
results committed to memory." After the general principles are 
understood, the accuracy of the illustrations may be tested by those 
who have time and taste for such calculations. A good computer 
who has learned the theory will then be able to detect and correct 
any arithmetical errors. 

There is nothing in the theory, the principles, or in the method' of 
calculating values in Life Insurance, that is not within the easy 
comprehension of men who have a thorough knowledge of the sin- 
gle rule of three, and a mere acquaintance with the simplest princi- 
ples of elementary algebra. The intelligent, general business sense 
of the country should be informed definitely what Life Insurance 
is, and what it is not ; and it is hoped that the following Notes may 
tend, in some degree, to promote this end. 

The rather complex equations given in the discussions for deter- 
mining the deposit on certain " return premium " policies may well 
be merely glanced over by those not directly interested in this pe- 
culiar kind of contract for insurance. The same may be said in re- 
gard to joint lives. But the principles used in constructing the 
commutation columns and the meaning of the values given in these 
tables should be closely studied. 



PAET I. -THE THEOEY. 



CHAPTER I. 

NET PREMIUMS. 

After a table of mortality and a rate of interest are designated, 
the method of calculating net values in life insurance is simple when 
carefully explained. The net premium is the amount that will, on 
the designated data — namely, rate of interest and table of mortality 
— exactly effect the insurance. In this calculation, expenses and 
profits arc left out of consideration. 

Life insurance calculations are made, first on the supposition that 
the amount insured is $1. Having obtained the net premium that 
will insure $1, the net premium that will effect similar insurance for 
$100, $1000, or any other named sum, is found by multiplying the 
net premium that will insure $1 by the sum actually insured. It is 
assumed that the payment of the premium is made at the time the 
insurance is effected ; this is called the beginning of the policy-year, 
and the amount insured is to be paid at the end of that policy-year, 
in which the insured may die. 

Interest. — The first thinsf to be considered, in making calculations 
for net premiums, is the method by which we determine the amount 
of money that will, when increased by interest at a given rate per 
annum, compounded annually, produce $1 in any designated number 
of years. Suppose that the rate of interest is 4 per cent per annum. 
The amount that will at this rate produce $1 in one year, when the 
principal and interest thereon for one year are added together, is ex- 
pressed by y.ij =={££ = $0.961538, plus other decimal figures. 

If the rate of interest is 4^ per cent per annum, the amount in 
hand that will, at this rate, produce $1 in one year is expressed by 
T y o = J-£™ = $0.956938. In general, divide one dollar by one plus 
the rate of interest in order to obtain the amount that will, if in- 
vested at this rate of interest, produce $1 in one year. 



14 NOTES ON LIFE INSURANCE. 

Having obtained the amount that will, at the given rate of inter- 
est, produce $1 in one year, in order to obtain the amount that will, 
at the same rate of interest per annum, produce $1 in two years, in- 
terest being compounded annually, we multiply the amount that will 
produce $1 in one year by itself. For instance, the amount that 
will produce $1 in one year at 4 per cent being expressed by ■^.^ ¥ , the 
amount that will, at the same rate of interest, compounded annually, 
produce $1 in two years, is expressed by T *-J ¥ x j.^ = $0.924556. 

If the rate of interest is 4J per cent, the expression becomes T .|^-g- 
X t.oVtt = $0.915730. 

The amount that will, at 4 per cent, produce $1 in three years is 
expressed by -gfa X T .Jx X y.fe = $0.888996. Interest being 4J 
percent, the expression becomes y.lVs" X ivfe" X T-inrs" = $0.876297. 

In general, to obtain the amount that will, if invested at any 
given rate of interest, compounded annually, produce $1 in any 
designated number of years, the rule is : Divide $1 by one plus the 
rate of interest, and multiply the quotient by itself a number of times 
equal to the number of years less one. That is to say, for two years, 
multiply once ; for three years, twice ; for four years, three times, 
and so on. 

These calculations have been made, and the results, at various 
rates of interest, have been placed in appended tables. 



Tables of Mortality. — There are at present two tables of mortality 
in quite general use in the United States. It is not claimed that 
either of them is in exact accordance with the rate of mortality 
among insured lives in this country. It is, however, maintained by 
life-insurance experts that either of the tables referred to is suffi- 
ciently accurate for practical purposes. The tables are given below. 
It will be noticed that in both it is assumed that there are 100,000 
persons living at age 10. In the column headed " number of 
deaths" will be found opposite age 10 the number that will die 
between age 10 and age 11. This will leave a certain number living 
at age 11. This number is placed opposite age 11, in the column 
headed "number living ;" and opposite age 11 in the column headed 
"number of deaths" is placed the number that die between age 11 
and age 12. 

For example, in the American Experience Table of Mortality, 
among insured lives, we find opposite age 10 in the column headed 
"number living," 100,000. Opposite same age we find, in the 
column headed " number of deaths," 749. This is the number, ac- 



NOTES ON LIFE INSURANCE. 



cording to this tabic, that will die between age 10 and age 11, and 
it will leave 90,251 living at age 11, of which number 740 will die 
between a Lie 11 and age 12. 



American Experience Table of Mortality. 



Age. 


Number living. 


Num. of death!?. 


' Age. 


Number living. 


Num. of deaths. 


10 


100,000 


749 


53 


66,797 


1091 


11 


5)9,251 


746 


54 


65,706 


1143 


12 


98,505 


713 


55 


64,563 


1199 


13 


97,762 


740 


56 


63,364 


1260 


14 


97,02-2 


73? 


57 


62,104 


1325 


15 


96/285 


735 


58 


60,779 


1394 


16 


95.550 


732 


59 


59,385 


1468 


17 


94,818 


729 


60 


57,917 


1516 


18 


94,089 


727 


61 


56,371 


1628 


19 


93*362 


725 


62 


54,743 


1713 


20 


92,637 


723 


63 


53,030 


1800 


21 


91,914 


722 


64 


51,230 


18S9 


22 


91,192 


721 


65 


49,341 


1980 


83 


90.471 


720 


66 


47,361 


2070 


24 


S9.751 


719 


67 . 


45,291 


2158 


25 


89,032 


718 


68 


43,133 


2243 


26 


88,314 


718 


69 


40,890 


2321 


27 


87.596 


718 


70 • 


38,569 


2391 


28 


86,878 


718 . 


71 


36,178 


2448 


29 


83,160 


719 : 


72 


33,730 


2487 


30 


85,441 


720 


73 


31,243 


2505 


31 


84,721 


721 


74 


28,738 


2501 


32 


84,000 


723 


75 


26,237 


2470 


33 


83,277 


726 


76 


23,761 


2431 


34 


82,551 


729 


77 


21,330 


2369 


85 


81.822 


732 


78 


18,961 


2291 


36 


81,090 


. 737 


79 


16,670 


2196 


87 


80.353 


742 


80 


14,474 


2091 


38 


79,611 


749 


81 


12,383 


1964 


39 


78,862 


756 


82 


10,419 


1S16 


40 


78,106 


765 


83 


8,603 


1648 


41 


77,341 


774 


84 


6,955 


1470 


42 


76,567 


735 


85 


5,485 


1292 


43 


75,782 


797 


86 


4,193 


1114 


44 


74,985 


812 


87 


3,079 


933 


45 


74,173 


828 


88 


2,146 


744 


46 


73.345 


848 


89 


1,402 


555 


47 


72,497 


870 


90 


847 


335 


48 


71,627 


896 


91 


462 


246 


49 


70,731 


927 


92 


216 


137 


50 


. 69,804 


962 


93 


■ -. 79 


58 


51 


68,842 


1001 


91 


. 21 


18 


52 


67,841 


1044 


95 


3 


3 



Actuaries' Table of Mortality. 



Age. 


Number living. 


Num. of deaths. 


Age. 


Number living. 


Num. of deaths. 


10 


100,000 


676 


21 


92,588 


6S3 


11 


93,324 


674 


22 


91,905 


686 


12 


98,650 


672 


23 


91,219 


690 


13 


97.978 


671 • 


24 


90,529 


694 


14 


97,307 


671 


25 


89,835 


693 


15 


96,636 


611 


26 


89.137 


703 


16 


95,965 


672 


27 


83,434 


708 


17 


95.293 


6T3 


28 


87,726 


714 


18 


94,620 


675 


29 


87.012 


720 


19 


93.945 


677 


30 


86.292 


727 


20 


93,268 


680 


31 


85,565 


731 



10 



NOTES OX LIFE INSURANCE. 
Actuaries' Table of Mortality — continued. 



Age 


dumber living. 


Num. of deaths. 


Age. 


Number living. 


Num. of deaths. 


32 


84,S31 


742 


66 


44.693 


2123 


33 


84.089 


750 


67 


42,565 


2191 


34 


83.339 


758 


68 


40,374 


2246 


33 


82.581 


767 


69 


38,128 


2291 


36 


81.814 


776 


70 


35,837 


2327 


37 


81.038 


785 


71 


33.510 


2351 


38 


80,253 


795 


72 


31,159 


2362 


39 


79.458 


805 


73 


28,797 


2358 


40 


78.653 


815 


74 


26,439 


2339 


41 


77,838 


826 


75 


24,100 


2303 


42 


77,012 


839 


76 


21,797 


2249 


43 


76,173 


857 


77 


19,548 


2179 


44 


75,316 


881 


78 


17,369 


2092 


45 


74.435 


909 


79 


15.277 


1987 


46 


73.526 


944 


80 


13.290 


1866 


47 


72,582 


981 


81 


11.424 


1730 


48 


71.601 


1021 


82 


9,694 


1582 


49 


70,580 


1063 


83 


8,112 


1427 


50 


69.517 


1108 


84 


6.685 


1268 


51 


68,409 


1156 


85 


5,417 


1111 


52 


67.253 


1207 


86 


4,306 


958 


53 


66,046 


1261 


87 


3,348 


811 


54 


64.785 


1316 


88 


2,537 


673 


55 


63.469 


1375 


89 


1,864 


545 


56 


62,094 


1436 


M) 


1,319 


427 


57 


60.658 


1497 


91 


892 


322 


58 


59:i61 


1561 


92 


570 


231 


59 


57.600 


1627 


93 


339 


155 


60 


55.973 


1698 


94 


184 


95 


61 


54.275 


1770 


95 


89 


52 


62 


52.505 


1844 


% 


37 


24 


63 


50.661 


1917 


97 


13 


9 


64 


48,744 


1993 


98 


4 


3 


05 


46,754 


2061 


99 


1 


1 



Various other tables of mortality are sometimes used in making 
life insurance calculations, but it is not necessary to give them here. 
Mortality tables are all based upon statistical information, obtained 
by observation and experience. It is assumed that each insured 
person at any particular age has the average chance of living for 
one year that is indicated by the table at that age. 

Manner of using the Mortality Table. — In illustration of the 
calculation of the chance that a person may die during any given 
time, let us suppose that out of one hundred persons condemned to 
be shot on a given day, all are reprieved for the day, except one, 
and the one to be shot is to be the one who draws the black ball out 
of a box containing ninety-nine white bails and one black one. The 
chance, before the drawing, of any particular man's getting the black 
ball, and, therefore, his chance of being shot that day is one only 
out of one hundred. If two men had to die that day, each indi- 
vidual's chance, before the drawing, of getting a black ball, would 
be twice as great as it was before ; for now there are two black 
balls and only ninety-eight white ones in the box. The chance, in 
this case, would be two out of one hundred ; for three persons, 



NOTES ON LIFE INSURANCE. 17 

three out of one hundred ; and so on to the limit of one hundred 
out of one hundred — which would make it certain that cacli man 
would be shot. 

To apply this principle to life insurance, and to show that the 
amount that will insure one dollar, to be paid certain at the end of 
any year, when multiplied by the fraction which represents the 
chance that the person will die during the year, gives what it is 
worth to insure one dollar, to be paid at the end of the year, in 
case the insured dies during the year : let us suppose that out of 
one hundred persons alive at the beginning of the year, it is known 
that one of them, and one only, will die during the year. The 
value of one dollar, to be paid certain at the end of one year, in 
the particular case of interest at four and one half per cent, is 
SO. 95 6 93 8. The chance that the person will die during the year is 
one out of one hundred. To make it certain that his heirs will obtain 
one dollar at the end of the year, he must advance $0.956938 at the 
beginning of the year. But the one dollar is not to be paid cer- 
tain ; it is to be paid only in case he dies. Suppose the whole one 
hundred persons are insured ; then, since there is but one to die, 
and but one dollar to be paid, each person will have to give only 
the one hundredth part of $0.956938 in order to make up what is 
necessary to pay this one dollar at the end of the year. Therefore, 
$0.956938 divided by 100 is what each man would have to pay. In 
case it is known that two persons out of the one hundred will die, 
the amount requisite to effect the insurance will be twice as much 
as before, because two dollars must be paid certain at the end of 
the year. The chance that each person may die during the year is 
twice as great as in the first case, and is therefore two out of one 
hundred ; and the amount that each person will have to pay for his 
insurance is $0.956938, divided by 100, and the result multiplied by 
two. In case it is known that all of them will die during the year, 
the fraction which represents the chance that any particular indi- 
vidual will die becomes ^-f^, or unity. This represents the certain- 
ty ; and to insure one dollar, to be paid at the end of the year to 
the heirs of the insured, in case he dies during the year, each per- 
son will now have to pay \%% of $0.956938 — that is to say, each 
must, in this case, pay enough to make, with interest at four and 
one half per cent added, the full amount for which he is insured. 

To insure One Dollar for One Year at Age 30. — The American 
Experience table shows that out of 100,000 persons living at age 
ten, there will be 85,441 living at age thirty. The number of 
deaths between age thirty and age thirty-one is 720. Therefore, 



18 



NOTES OX LIFE INSURANCE. 



"5" 544T ^ S tne fraction which represents the chance that the insured 
will die before he is thirty-one years of age. 

The present value of one dollar to be paid certain at the end of 
one year has, in the case of interest at four and one half per cent, 
been found to be equal to 80.956938. Multiply this by the fraction 
8 ||2 1 1 which represents the chance that the insured will die during 
the year between age thirty and age thirty-one, and we have 
$0.0081064. This is the value of the risk on one dollar, or what it 
is worth to insure one dollar to be paid to the heirs of a person at 
the end of one year, in case he dies during the year, the age of the 
insured being thirty years at the time he takes out the policy. It 
would require one thousand times as much to insure one thousand 
dollars as it does to insure one dollar, and half as much to insure 
half a dollar as it does a whole dollar. We can obtain, by using 
the mortality table, the fraction representing the chance that a per- 
son of any given age will die during the succeeding year, just as we 
-did in this case for age thirty. We have, therefore, already deter- 
mined the means for calculating the sum that will at any age insure 
one dollar to be paid to the heirs of the insured in one year, in case 
he dies during the year ; but we must know the age of the person, 
the rate of interest must be fixed, and a mortality table must be 
available for use in the calculations. 

The following table shows the net amount that will insure $1000 
for one year at different ages from 20 to Y0 inclusive, the calcula- 
tions being all based upon the American Experience Table of Mor- 
tality, b)ut .at different rates of interest, 4, £}, 5, and 6 per cent. 



Cost of Insurance on $1000, for one year, at different Ages- 
American Experience — Various Rates of Interest. 



Age. 


Tour j>er cent 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


20 


$7,504 


$7,469 


$7,433 


$7,363 


20 


21 


7.553 


7.517 


7 481 


7.411 


21 


22 


7.602 


7.566 


7.530 


7.459 


22 


23 


7.652 


7.616 


7.579 


7.508 


53 


24 


7.703 


7.666 


7.630 


7.558 


24 


25 


7.754 


7.717 


7.680 


7.608 


25 


26 


7.815 


7.780 


7.743 


7.670 


26 


27 


7.881 


7. £44 


7.806 


7.733 


27 


28 


7.947 


7.909 


7.871 


7.797 


28 


3& 


8.024 


7-986 


7.948 


7.873 


29 


30 


8.103 


8.064 


8.026 


7.950 


30 


:31 


8.183 


8.144 


8.105 


8.029 


31 


32 


8.276 


8.237 


8.197 


8.120 


32 


33 


8.383 


8.342 


8.303 


8.224 


33 


34 


8.491 


8.451 


8.410 


8.331 


34 


35 


8.602 


8.561 


8.520 


8.440 


35 


36 


8.739 


8.697 


8.656 


8.574 


36 


37 


8.879 


8.837 


8.795 


8.712 


37 


38 


9.046 


.9.003 


8.960 


8.876 


38 



NOTES ON LIFE INSURANCE. 



19 



Cost of Insurance on $1000 — continued 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


39 


$9,218 


$9,174 


$9,130 


$9,044 


39 


40 


9.418 


9.373 


9.328 


9.240 


40 


41 


9.623 


9.577 


9.531 


9.441 


41 


42 


9.858 


9.811 


9.764 


9.672 


42 


43 


10.113 


10.064 


10.016 


9.922 


43 


44 


10.412 


10.363 


10.313 


10.216 


44 


45 


10.734 


10.682 


10.632 


10.531 


45 


46 


11.117 


11.064 


11.011 


10.907 


46 


47 


11.539 


11.484 


11.429 


11.321 


47 


48 


12.028 


11.971 


11.914 


11.801 


48 


49 


12.602 


12.542 


12.482 


12.364 


49 


50 


13.251 


13.188 


13.125 


13.001 


50 


51 


13.981 


13.914 


13.848 


13.717 


51 


52 


14.797 


14.726 


14.656 


14.518 


52 


53 


15.705 


15.629 


15.555 


15.409 


53 


54 


16.727 


16.647 


16.567 


16.411 


54 


55 


17.857 


17.771 


17.687 


17.520 


55 


56 


19.120 


19.029 


18.938 


18.760 


56 


57 


20.515 


20 416 


20.319 


20.128 


57 


58 


22.053 


21.948 


21.843 


21.637 


58 


59 


23.769 


23.656 


23.543 


23.321 


59 


60 


25.667 


25.544 


25.422 


25.182 


60 


61 


27.769 


27.636 


27.503 


27.245 


61 


62 


30.088 


29.944 


29.802 


29.520 


62 


63 


32.638 


32.481 


32.327 


32.022 


63 


64 


35.455 


35.285 


35.117 


34.786 


64 


65 


38.585 


38.401 


38.218 


37.857 


65 


66 


42.026 


41.825 


41.626 


41.233 


66 


67 


45.815 


45.596 


45.379 


44.950 


07 


68 


50.002 


49.763 


49.526 


49.058 


68 


69 


54.579 


54.318 


54.060 


53.549 


69 


70 


59.608 


59.323 


59.041 


58.484 


70 



For the purpose of comparison, the following table is inserted : 



Cost of Insurance on $1000, for one year, at different Ages — Actu- 
aries'' Table — Various Hates of Interest. 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


20 


$7,010 


$6,977 


$6,944 


$6,878 


20 


21 


7.093 


7.059 


7.026 


6.959 


21 


22 


7.177 


7.143 


7.109 


7.042 


22 


23 


7.273 


7.238 


7.204 


7.136 


23 


24 


7.371 


7.336 


7.301 


7.232 


24 


25 


7.471 


7.435 


7.400 


7.330 


25 


26 


7.583 


7.547 


7.511 


7.440 


26 


27 


7.698 


7.661 


7.625 


7.553 


27 


28 


7.826 


7.788 


7.751 


7.678 


28 


29 


7.956 


7.918 


7.881 


7.807 


29 


30 


8.101 


8.062 


8.024 


7.948 


30 


31 


8.248 


8.209 


8.170 


8.093 


31 


32 


8.410 


8.370 


8.330 


8.252 


32 


33 


8.576 


8.535 


8.494 


8.414 


33 


34 


8.746 


8.704 


8.662 


8.581 


34 


35 


8.931 


8.888 


8.845 


8.762 


35 


36 


9.122 


9.076 


9.033 


8.948 


36 


37 


9.314 


9.270 


9.225 


9.138 


?7 


38 


9.525 


9.480 


9.435 


9.346 


38 


39 


9.741 


9.695 


9.649 


9.558 


39 


40 


9.963 


9.916 


9.869 


9.775 


40 


41 


10.204 


10.155 


10.106 


10.011 


41 


42 


10.476 


10.425 


10.375 


10.278 


42 


43 


10.818 


10.766 


10.715 


10.614 


43 


44 


11.247 


11.194 


11.140 


11.035 


44 



20 



NOTES ON LIFE INSURANCE. 



Cost of Insurance on $1000 — continued. 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


45 


$11,742 


$11,686 


$11,630 


$11,521 


45 


46 


12.345 


12.286 


12.227 


32.112 


46 


47 


12.996 


12.934 


12.872 


12.751 


47 


48 


13.711 


13.646 


13.580 


13 452 


48 


49 


14.482 


14.412 


14.344 


14.209 


49 


60 


15.326 


15.252 


15.180 


15.036 


50 


51 


16.248 


16.171 


16.093 


15.941 


51 


52 


17.257 


17.174 


17.093 


16.931 


52 


53 


18 359 


18.271 


18.183 


18.012 


53 


54 


19.532 


19.439 


19.346 


19.163 


54 


55 


20.831 


20.731 


20.633 


20.438 


55 


56 


22.237 


22.130 


22.025 


21.817 


56 


57 


23.730 


23.617 


23.504 


23.282 


57 


58 


25.371 


25.249 


25.129 


24.892 


58 


59 


27.160 


27.080 


26.901 


26.647 


59 


60 


29.169 


29.030 


28.891 


28.619 


60 


61 


31.357 


31.207 


31.059 


30.765 


61 


62 


33.770 


33.608 


33.448 


33.132 


62 


63 


36.384 


36.210 


36.038 


35.698 


63 


64 


39.255 


39.068 


38.882 


38.515 


64 


65 


42.386 


42.184 


41.983 


41.586 


65 


66 


45.782 


45.563 


45.346 


44.919 


66 


67 


49.494 


49.258 


49.023 


48.560 


67 


68 


53.490 


53.234 


52.981 


52.481 


68 


69 


57.776 


57.500 


57.226 


56.686 


69 


70 


62.436 


62.137 


61.841 


61.257 


70 



In further illustration of the question of net premiums and net 
values of life policies in case the insurance is for one year only ; 
we will suppose that the age of the person applying for insurance is 
40, the mortality table used the Actuaries', the rate of interest is 
four per cent, the insurance to be for one year, the amount of the 
policy $10,000, and the policy-holder is a flair average of insured 
lives. What amount in hand ought to be paid to insure $10,000 as 
above, leaving out all consideration of expenses, and having neither 
gain nor loss represented in the chances of the transaction ? 

We will first determine the amount that will insure $1. The 
present value of $1, to be paid certain at the end of one year, 
interest being assumed at four per cent per annum, is equal to 
$0.961538. From the mortality table we find that out of 100,000 
persons living at age 10, there are 78,653 living at age 40, and that 
815 of these will die during the year between age 40 and age 41. 
Therefore, ^g| j a is the fraction which represents the chance that 
the insured will die during the year ; and this multiplied by 
$0.961538, which is the present value of $1, to be paid certain at 
the end of one year, will give what it is now worth to insure the $1, 
to be paid in case the insured dies during the year, or $0.961538 
X T |i-|^=$0.009963432. This, multiplied by 10,000, makes $99.63, 
plus a fraction of a cent, which is the net premium that will insure 



NOTES ON LIFE INSURANCE. 21 

$10,000, to be paid to the heirs of the person insured at the end of 
one year ; provided he dies during the year. 

If every one of the whole 78,653 had been insured for one year 
in the sum of $10,000 each, the heirs of the 815 who died were re- 
spectively entitled to, and were paid, $10,000 each. The aggregate 
payment of losses by death, for the year, amounted, therefore, to 
$8,150,000. The net annual premiums, $99.63 each, on 78,653 poli- 
cies, when increased by four per cent interest, amounted at the end 
of the year to $8,149,996.43. 

It is seen from this, that the fraction of a cent omitted in each 
of the 78,653 premiums amounted, in the aggregate, to a deficiency 
of $3.57, in paying $8,150,000 losses by death. The 77,838 persons 
who did not die during the year, in this case receive nothing. They 
are not entitled to any thing ; they had their lives insured during 
the year ; they each paid in advance a net premium amounting to 
$99.63 ; the whole of this, and net interest on it, has gone to pay 
at the end of the year $10,000 to the heirs of each of the 815 policy- 
holders that died during the year. 

Net Single Premium for Insurance for Whole Life. — What sum 
paid in hand will, " on the supposition that the mortality table and 
rate of interest designated as the basis of the calculations are cor- 
rect" be the exact equivalent of $1 insured, to be paid at the end 
of any year in which the insured may die ? In other words, we 
have now to determine the amount that, if paid in hand, will exact- 
ly insure $1 for whole life. The problem before us reduces to this, 
namely : First, calculate the present value of the amount requisite 
to effect the insurance each separate year that the insured may live, 
as indicated by the designated table of mortality. Then add to- 
gether the respective amounts that are found to be necessary to 
effect the insurance each year, and their sum will give the amount 
that, if paid in hand, will effect the insurance for whole life. 

Assume that the age of the person at the time he insures is 40, 
that he is a fair average of insured lives, the table of mortality is 
the Actuaries', and the rate of interest four per cent, compounded 
annually. We have already found that the amount necessary in 
this case to insure $1 to be paid to the heirs of the insured in case 
he dies during the first year — that is, between age 40 and age 41 — is 
$0.009963432. Now, let us see what amount of money, if paid in 
hand at the time the policy is issued (age 40), will secure $1 to be 
paid to the heirs of the insured at the end of two years, in case he 
dies in the second year from the date of the policy. This problem 



22 NOTES ON LIFE INSUKANCE. 

is solved separately, and the amount we are now to find insures 
only against death in the second year. 

Find the amount that will, if invested at 4 per cent per annum, 
compound interest, produce $1 in two years. To do this, we have 
to divide 100 by 104 ; this gives $0.961538, and this is the amount 
that will produce $1 in one year at 4 per cent per annum. Multiply 
this by itself, and we have $0.924556, which is the amount that will 
produce $1 in two years at 4 per cent compounded annually. 

Now, the question is, "What fraction, at age 40 (the time at which 
the insurance is effected), represents the chance that the insured 
will die during the second year — that is, between age 41 and age 
42? 

By the Actuaries' Table of Mortality, we see that out of 78,653 
insured persons living at age 40, the number that will die between 
age 41 and age 42 is 826. Therefore, the fraction 826 divided 
by 78,653 represents at the time this insurance is effected, 
age 40, the chance that the insured will die between age 41 and 
age 42. 

Multiply the amount, $0.924556, that will, at the designated rate 
of interest, produce $1 certain in two years, by the fraction 7 ||| 3 , 
which represents the chance that the $1 insured will have to be 
paid, and we have $0.009705, which is the amount that will, if paid 
in hand at age 40, insure $1 to be paid to the heirs of the insured in 
case he dies in the second year. 

In like manner, we can find the amount that, if paid in hand at 
age 40, will insure $1 to the heirs of the insured in case he dies in 
the third year — and for each and every year up to and including the 
table limit. 

Add together these respective yearly amounts, and the result 
gives the net single premium that will, if naid in hand at age 40, 
insure $1 for whole life. 

The amount in this case is $0.38104 ; multiply this by 1000, and 
we have $381.04, which is the net single premium that will at age 
40 insure $1000, to be paid to the heirs of the insured at the end of 
any year in which he may die. 

Detailed Calculation of Net Single Premium for Whole-Life 
Policies. — In illustration of this subject, we will assume that the 
age of the insured is 50. The amount insured is $1. The table of 
mortality used is the American Experience, and the rate of interest 
is 4^ per cent per annum, compounded yearly. Opposite age 50 in 
the following table, in the column headed, " Value in hand at age 



NOTES ON LIFE INSURANCE. 23 

50 of $1 to be paid certain at the end of each respective year" we 
find the amount $0.956938. This is the amount that at 4£ per cent 
will produce $1 in one year, and it is obtained by dividing 100 by 
104£. This is to be multiplied by the number of deaths during the 
year between age 50 and age 51 ; this number, as shown by the 
mortality table, is 962, and it is placed in the following table oppo- 
site age 50, in the column headed, "Number of Deaths.'' In the 
next column, which is headed, " Number Living at Age 50," we 
find the number 69804 ; this, too, is taken from the table of mor- 
tality. The product arising from multiplying $0.956938 by 962 is 
divided by 69804, and the result is $0.013188. This is the amount 
that at age 50 will insure $1, to be paid to the heirs of the insured 
at the end of the year, in case he dies between age 50 and age 51. 

Opposite age 51, in the same table, in the column headed, 
" Value in hand at age 50 of $1, to be paid certain at the end of 
each respective year," we find the amount $0.915730. This is the 
amount that will produce $1 certain in two years, when invested at 
4J per cent per annum, compounded annually, and it was obtained 
by multiplying T .^j by y.^j. Multiplying this by 1001, which is 
the number of deaths between age 51 and age 52, and divide the 
product by 69804, which is the number living at age 50, and we 
have $0.013132. This is the amount that, if paid in hand at age 50, 
will insure $1, to be paid to the heirs of the insured at the end of 
the second year, in case the insured dies in that year — that is, be- 
tween age 51 and age 52. 

In a similar manner, the calculations as shown in the table are 
made each respective year to include age 95, which is the limit of 
the table of mortality we are now using, and the sum of all these 
respective yearly values gives us $0.430037. This amount, if paid 
in hand at age 50, will insure $1 for whole life on the data assumed 
— namely, the American Experience Table of Mortality, and 4^ per 
cent per annum, compounded yearly. 

In an entirely similar manner, the calculations can be made at 
any age included in the table of mortality, and at any designated 
rate of interest. 

Having found the net single premium that will insure $1 at any 
age, the net single premium that will insure any other amount at 
the same age is found by a simple proportion. 



24 



NOTES ON LIFE INSURANCE. 



Table. — Illustrating the manner of calculating the net Single JPre- 
miumfor a Whole-Life Policy for $1 ; issued at age 50 — Ameri- 
can Experience, four and a half per cent. 



Age. 


Value in hand, at age 
50, of SI, to be paid cer- 
tain at the end of each, 
respective year. 


Number of 
Deaths. 


Nuraber living 
at age 50. 


Value, in band, at age 50, 
of $1, to be paid at the 
end of each respective 
year, provided the insured 
dies duriug the year. 


Age. 


50 


$0.956938 


X 


962 -f- 


69804 




$0.013188 


50 


51 


0.915730 


X 


1001 


4- 


69804 




.013132 


51 


52 


0.876297 


X 


1044 


4- 


69804 




.013106 


52 


53 


0.838561 


X 


1091 


+- 


69804 




.013106 


53 


54 


0.802451 


X 


1143 




69804 


= 


.013140 


54 


55 


0.767896 


X 


1199 




69804 




.013190 


55 • 


56 


0.734828 


X 


1260 


4- 


69804 




.013264 


56 


57 


0.703185 


X 


1325 


4- 


69804 




.013348 


57 


58 


0.672904 


X 


1394 


4- 


69804 




.013438 


58 


59 


643928 


X 


1468 


4- 


69804 




.013542 


59 


60 


0.616199 


X 


1546 




69804 




.013647 


60 


61 


0.589664 


X 


1628 


4- 


69804 




.013752 


61 


62 


0.564272 


X 


1713 


4- 


69804 




.013847 


62 


63 


0.539973 


X 


1800 -5- 


69804 




.013924 


63 


64 


0516720 


X 


1889 -4- 


69804 




.013983 


64 


65 


0.494469 


X 


1980 -*- 


69804 




.014026 


65 


66 


0.473176 


X 


2070 -*- 


69804 




.014032 


66 


67 


0.452800 


X 


2158 ■*- 


69804 




.013998 


67 


68 


0.433302 


X 


2243 -*- 


69804 




.013923 


68 


69 


0.414643 


X 


2321 


4- 


69804 




.013787 


69 


70 


0.396787 


X 


2391 -*- 


69804 




.013591 


70 


71 


0.379701 


X 


2448 -r- 


69804 




.013316 


71 


72 


0.363350 


X 


2487 -*- 


69804 




.012946 


72 


73 


0.347703 


X 


2505 -*- 


69804 




.012478 


73 


74 


0.332731 


X 


2501 




69804 




.011921 


74 


75 


0.318402 


X 


2476 -f- 


69804 




.011294 


75 


76 


0.304691 


X 


2431 -*- 


69804 




.010611 


76 


77 


0.291571 


X 


2369 -?- 


69804 




.009895 


77 


78 


0.279015 


X 


2291 -4- 


69804 




.009157 


78 


79 


0.267000 


X 


2196 


4- 


69804 




.008400 


79 


80 


0.255502 


X 


2091 


4- 


69804 




.007654 


80 


81 


0.244500 


X 


1964 


4- 


69804 




.006879 


81 


82 


0.233971 


X 


1816 


4- 


69804 




.006087 


82 


83 


0.223896 


X 


1648 -f- 


69804 




.005286 


&3 


84 


0.214254 


X 


1470 -*- 


69804 




.004512 


84 


85 


0.205028 


X 


1292 -*- 


69804 




.003.95 


85 


86 


0.196199 


X 


1114 -5- 


69804 




.003131 


86 


87 


0.1S7750 


X 


933 -4- 


69804 




.002509 


87 


88 


0.179665 


X 


744 -*- 


69804 




.001915 


88 


89 


0.171929 


X 


555 




69804 




.001367 


89 


90 


0.164525 


X 


385 -4- 


69804 




.000907 


90 


91 


0.157440 


X 


246 -*- 


69804 




.000555 


91 


92 


0.1 0661 


X 


137 -*- 


69804 




.000296 


92 


93 


0.144173 


X 


58 - 




G9804 




.000120 
.000036 


93 


94 


0.137964 


X 


18 -*- 


69804 




94 


95 


0.132023 


X 


3 + 


69804 




.000006 


95 




Total 












..$0.430037 





















If the insurance is for a limited number of years only, the calcu- 
lation is made for each year of the term, and the sum of these year- 
ly amounts will give the net single premium that will, if paid at the 
time the policy is issued, insure %\ to be paid to the heirs of the in- 
sured at the end of the year in which the policy-holder may die ; 
provided he dies before the expiration of the term. For instance, 
suppose the insurance at age 50 is to continue for ten years only ; 



NOTES ON LIFE INSURANCE. 



25 



take in the above table the amounts for the first ten years, their sum 
will be the net single premium required. 

If insurance is paid for at age 50, to begin only at the end of ten 
years from that time, and then continue for life ; the net single pre- 
mium is obtained by subtracting the amount that will at age 50 in- 
sure $1 until age 60, from the amount that will at age 50 insure $1 
for life. 

If insurance is paid for at age 50 to begin at age GO, and then con- 
tinue for ten years only : subtract the net single premium that will 
at age 50 insure $1, to begin at age 10 and continue for life, from 
the net single premium that will at age 50 insure $1, to begin at age 
60 and continue for life. 

The net single premium that will at any age insure $1, to be paid 
at the end of any designated year, provided the insured dies in that 
year, is found, as before stated, by first determining the amount 
that will at the named rate of interest produce $1 at the end of the 
designated year, and then multiplying this amount by the fraction 
(obtained from the mortality table), which represents, at the time 
the insurance is effected, the chance that the insured will die in the 
designated year. 

For purposes of comparison and general illustration, the following 
tables are given, showing the net single premiums that will insure 
$1000 for whole life, at ages from 20 to 70 inclusive, at 4, 4-J, 5, and 
6 per cent respectively: 



Net Single Premiums — American Experience- 
Interest. 



Various Hates of 



Age 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


20 


247.798 


217. 44S 


192.545 


154.766 


20 


21 


251.846 


221.155 


195.896 


157.476 


21 


22 


256.076 


225.019 


199.402 


160.329 


22 


23 


260.472 


229.049 


203.071 


163 334 


23 


24 


265.042 


233.255 


206.913 


166.501 


24 


25 


269.704 


237.644 


210.937 


169.840 


25 


26 


274.737 


242.227 


215.155 


173.364 


26 


27 


279.872 


247.005 


219.568 


177.075 


27 


28 


285.207 


251.989 


224.187 


180.987 


28 


29 


290.754 


257.189 


229.025 


185.111 


29 


30 


296.514 


262.609 


234.084 


189.454 


80 


31 


302.497 


268.261 


239.378 


194.029 


31 


32 


308.713 


274.155 


244.922 


198.853 


32 


33 


315.167 


280.298 


250.718 


203.933 


33 


34 


321.862 


286.692 


'256.775 


209.275 


34 


35 


328.809 


293.353 


263.107 


214.898 


35 


36 


336.022 


300.294 


269.728 


220.822 


36 


37 


343. 08S 


307.514 


276.552 


227.046 


37 


38 


351.245 


315.027 


283.859 


233.591 


38 


39 


359.266 


322.832 


291.385 


240.458 


39 


40 


367.575 


3 0.946 


299.237 


247.676 


40 


41 


376.167 


339.363 


308.371 


255.242 


41 



26 



NOTES ON LIFE INSURANCE. 
Net Single Premiums— continued 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


42 


3S5.060 


348.115 


315.940 


263.183 


42 


43 


394.252 


357.190 


324.815 


271.505 


43 


44 


403.751 


366.602 


334.051 


280 225 


44 


45 


413.551 


376.346 


343.646 


289.343 


45 


46 


423.659 


386.432 


353.613 


298.877 


46 


47 


434.063 


396.848 


363.939 


308.818 


47 


48 


444.762 


407.597 


374.632 


319.178 


48 


49 


455.744 


418.667 


385.679 


329.947 


49 


50 


467.046 


430.037 


397.060 


341.108 


50 


51 


478.480 


441.695 


408.767 


352.653 


51 


52 


490.207 


453.626 


420.781 


364.572 


52 


53 


502.154 


465.819 


433.096 


376.857 


53 


54 


514.307 


478.259 


445.698 


389.497 


54 


55 


526.645 


490.925 


458.564 


402.473 


55 


56 


539.153 


503.802 


471.681 


415.771 


56 


57 


551.806 


516.866 


485.024 


429.371 


57 


58 


564.589 


530.099 


498.578 


443.255 


58 


59 


577.482 


543.484 


512.321 


457.405 


59 


60 


590.457 


556.989 


526.226 


471.792 


60 


61 


603.491 


570.591 


540.266 


486.390 


61 


62 


616.557 


584.261 


553.566 


501.166 


62 


63 


629.630 


597.973 


568.631 


516.094 


63 


64 


642.687 


611.702 


582.905 


531.143 


64 


65 


655.699 


625.416 


597.198 


546.284 


65 


66 


668.630 


639.076 


611.467 


561.464 


66 


67 


681.452 


652.653 


625.679 


576.648 


67 


68 


694.137 


666.114 


639.801 


591.797 


68 


69 


706.647 


679.418 


653.787 


606.861 


69 


70 


718.960 


692.540 


667.610 


621.805 


70 



JVet Single Premiums — Actuaries'' Table — Various Pates of Interest. 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


20 


251 .907 


221.069 


195.651 


156.926 


20 


21 


256.564 


225.373 


199.598 


160.219 


21 


22 


261.377 


229.830 


203.703 


163.662 


22 


23 


266.357 


234.458 


207.977 


167.236 


23 


24 


271.500 


239.254 


212.418 


171.032 


24 


25 


276.816 


244.226 


217.037 


174.969 


25 


26 


282.312 


249.384 


221.843 


179.089 


26 


2" 


287.990 


254.728 


226.837 


183.394 


27 


28 


293.856 


260.269 


232.030 


187.896 


28 


29 


299.913 


266.007 


237.425 


192.598 


29 


30 


306.168 


271.952 


243.033 


197.513 


30 


31 


312.624 


278.108 


248.856 


202.646 


31 


32 


319.289 


284.485 


254.907 


208.011 


32 


33 


326.167 


291.086 


261.191 


213.614 


33 


34 


333.267 


297.922 


267.719 


219.469 


34 


35 


340.600 


304.443 


274.506 


225.593 


35 


36 


348.170 


312.346 


281.559 


231.996 


36 


37 


355.989 


319.951 


2S8.892 


238.695 


37 


38 


364.065 


327 838 


296.522 


245.710 


38 


39 


372.414 


336.012 


304.458 


253.053 


39 


40 


381.040 


344.491 


312.718 


260.747 


40 


41 


389.960 


353.292 


321.321 


268.815 


41 


42 


399.183 


363.425 


330 283 


277.275 


42 


43 


408.709 


371.891 


339.600 


286 134 


43 


44 


418.515 


381.668 


349.258 


295.374 


44 


45 


428.571 


391.729 


359.226 


304.967 


45 


46 


438.862 


402.054 


369.487 


314.899 


46 


47 


449.346 


412.604 


380.002 


325.128 


47 


48 


460.022 


423.390 


390.767 


335.656 


48 



NOTES ON LIFE INSURANCE. 
Net Single Premiums — continued. 



27 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


49 


470.878 


434.864 


401.775 


346.477 


49 


50 


481.906 


445.560 


413.024 


357.590 


50 


51 


493.107 


456.955 


424.502 


368.983 


51 


52 


504.460 


468.537 


436.200 


380.661 


52 


53 


515.949 


480.293 


448.105 


392.600 


53 


54 


527 567 


492 211 


460.204 


404.792 


54 


55 


539.312 


504.291 


472.498 


417.241 


55 


56 


551.157 


516.509 


484.966 


429.926 


56 


57 


563.103 


528.856 


497.596 


442.836 


57 


58 


575.142 


541.335 


510.392 


455.980 


58 


59 


587.257 


553.925 


523.335 


469.337 


59 


60 


599.433 


566.610 


536.407 


482.891 


60 


61 


611.628 


579.346 


549.564 


496.593 


61 


62 


623.826 


592.115 


562.783 


510.423 


62 


63 


635.995 


604.883 


576 032 


524.343 


63 


64 


648.120 


617.633 


589.292 


538.334 


64 


65 


660.171 


630.835 


602.530 


552.359 


65 


67 


672.124 


642.961 


615.716 


566.386 


66 


67 


683 968 


655.491 


628.830 


580 390 


67 


68 


695.654 


667.892 


641.835 


594.332 


68 


69 


707.192 


680.154 


654.713 


608.196 


69 


70 


718.569 


692.271 


667.474 


621.973 


70 



Net Annual Premium for Whole-Life Policies. — Having shown 
how to calculate the net single premium that will insure one dollar 
for whole life at any age included in the table of mortality, it is now 
proposed to see how the exact equivalent can be determined when 
the payments are made by installments instead of one sum in ad- 
vance. It is usual, in ordinary whole-life policies, for the compa- 
nies to charge, and the insured to pay, equal annual premiums, the 
first in advance, and one at the beginning of each following year, as 
long as the insured is alive. When the net single premium at any 
age has been accurately calculated, the question arises, What is the 
net annual premium, under the conditions just named, that will be 
the precise equivalent of this net single premium at the time the 
policy issued? 

When this net annual premium is accurately determined, it being 
the precise money equivalent of the net single premium, and the net 
single premium being just sufficient, on the data designated, to effect 
the insurance, it follows that its precise equivalent, paid in equal 
annual premiums, will also be just sufficient to effect the insurance. 

To solve this problem, first find the value, at any named age, of 
a whole-life series of annual premiums of $1 each, the condition 
being that the first payment of $1 is to be made at the time the 
policy is issued, and that $1 is to be paid at the beginning of each 
following year, as long as the person is alive to make the payment. 
When this is done, the problem we wish to solve becomes simple, 
because, after we have found, at any named age, the value in hand 



28 NOTES ON LIFE INSURANCE. 

of this life series of premiums of #1 each, this amount will be to the 
net single premium at that age as $1 is to the annual premium re- 
quired. "We will use the American Experience Table of Mortality 
and 44 per cent interest. 

The first thing, then, is to find the value at any age — say 50 — of 
a life series of annual premiums of $1 each, the first to be paid in 
advance, and $1 to be paid at the beginning of each following year, 
as long as the insured is alive to make the payments. 

The first annual payment is to be made in hand, and its value, at 
the time the policy is issued, is $1. The value in hand of 81, to be 
paid at the end of one year, interest being 4| per cent per annum, is 
100 divided by 104^ ; but this second payment is not to be made 
certain, but only on condition that the insured, aged 50, will be alive 
at age 51. The fraction which, at age 50, represents the chance that 
the insured will be alive at age 51, is equal to the number living at 
age 51, divided by the number living at age 50. From the table of 
mortality, we find the number living at age 51 is 68,824, and the 
number living at age 50 is 69,804. The fraction in this case is, 
therefore, f-ffff. The value, at age 50, of this second payment of 
$1 to be made at age 51, in case the insured is alive to make the 
payment, is equal to jtoVs X SSI oa* -"- n ^^ e manner, the value of 
the third payment, which is to be made at the end of two years, in 
case the insured is alive at that time to make the payment, is equal 
t0 it^it X \-.-oTTi multiplied by the fraction which, at age 50. repre- 
sents the chance at that time that the insured will be alive at age 
52 to make the third payment. This fraction is equal to the num- 
ber living at 52, divided by the number living at 50, which we see, 
from the table of mortality, is flfii 5 therefore the value in hand, 
at age 50, of the third payment of $1, to be paid only on condition 
that the insured is alive at age 52, is expressed by -^oTs X t-oV? X 

6_5841 
6 9804* 

In a manner entirely similar, we can calculate at age 50 the value 
in hand of the fourth and every payment of the whole-life series to 
the limit of the table. This has been done, and the sum of the re- 
spective values in hand, at age 50, of the whole-life series of annual 
premiums of $1 each, is found to be 813.235S02. 



NOTES ON LIFE INSURANCE. 



29 



Table, illustrating the manner of calculating the value, at age 50, of 
a Whole-Life Series of Annual Payments of $1 each — the first 
payment to be made in hand, and one at the beginning of each fol- 
lowing year, as long as the perso?i is alive to make the payment. 
American Experience— four and a half per cent. 















Value in hand, at age 




Age. 


Value in hand, at age 
50, of $1, to be paid 
certain at the begin- 
ning of each year. 


Number liv- 
ing at be- 
ginning of 
each year. 

Numerator. 


Number living 

at age 50. 
Denominator. 




50, of $1, to be paid 
at the beginning of 
each respective year, 
provided the person 
is alive to malce the 


Age. 
















payment. 




50 


$1.000000 


X 


69804 


. 


69804 


_ 


$1.000000 


50 


51 


0.956938 


X 


68842 


-4- 


69804 


= 


0.9^3750 


51 


52 


0.915730 


X 


67841 


■+■ 


69804 


= 


0.889978 


52 


53 


0.876297 


X 


66797 


-i- 


69804 


= 


0.838548 


53 


54 


0.838561 


X 


65706 


-|- 


- 69304 


— 


0.789331 


54 


55 


0.802451 


X 


64563 


-J- 


69804 


= 


0.742202 


55 


56 


0.767896 


X 


63364 


-4- 


69804 


= 


0.697051 


56 


57 


0.734828 


X 


62104 


-J- 


69804 


= 


0.653770 


57 


58 


0.703185 


X 


60779 


-¥• 


69804 


— 


0.612270 


58 


59 


0.672904 


X 


59385 


■+■ 


69804 


= 


0.572466 


59 


60 


0.643928 


X 


57917 


■+■ 


69804 


= 


0.534273 


60 


61 


0.616199 


X 


56371 


-*• 


69804 


S3 


0.497619 


61 


ea 


0.589664 


X 


54743 


■+• 


69804 


33 


0.462437 


62 


63 


0.564272 


X 


53030 


-*- 


69804 


— 


0.428677 


63 


64 


0.539973 


X 


51230 


-f- 


69804 


S3 


0.396293 


64 


65 


0.516720 


X 


49341 


-*• 


69804 


S3 


0.365244 


65 


66 


0.494469 


X 


47361 


-*- 


69804 


S3 


0.335490 


66 


67 


0.473176 


X 


45291 


-*- 


69804 


33 


0.307011 


67 


68 


0.452800 


X 


43133 


-> 


69804 


S3 


0.279792 


68 


69 


0.433302 


X 


40890 


-*- 


69804 


— 


0.253821 


69 


70 


0.414643 


X 


38569 


-f- 


69804 


— 


0.229104 


70 


71 


0.396787 


X 


36178 


-J- 


69304 


S= 


0.205647 


71 


72 


0.379701 


X 


33730 


-*- 


69804 


S3 


0.183475 


72 


73 


0.363350 


X 


31243 


-J- 


69804 


S3 


0.162629 


73 


74 


0.347703 


X 


28738 


-*• 


69804 


S3 


0.143148 


74 


75 


0.332731 


X 


26237 


■+■ 


69804 


S3 


0.125063 


75 


76 


0.318402 


X 


23761 


+ 


69804 


— 


0.108383 


76 


77 


0.304691 


X 


21330 


-*- 


69804 


= 


0.093104 


77 


78 


0.291571 


X 


18961 


-*- 


69804 


S3 


0.079200 


78 


79 


0.279015 


X 


16670 


■+■ 


69804 


S3 


0.066632 


79 


80 


0.267000 


X 


14474 


-*- 


69804 


S3 


0.055363 


80 


81 


0.255502 


X 


12383 


•*■ 


69804 


S3 


0.045325 


81 


82 


0.244500 


X 


10419 


-i- 


69804 


— 


0.036494 


82 


as 


0.233971 


X 


8603 


-*- 


69804 


S3 


0.028836 


83 


84 


0.22:3896 


X 


6955 


-T- 


69804 


= 


0.022308 


84 


85 


0.214254 


X 


5485 


-*- 


69804 


33 


0.016835 


85 


86 


0.205023 


X 


4193 


-*- 


69804 


3S 


0.012316 


86 


87 


0.196199 


X 


3079 


-J- 


69804 


=3 


0.008654 


87 


88 


0.187750 


X 


2146 


-f- 


69804 


S3 


0.005772 


88 


89 


0.179665 


X 


il402 


-J- 


69804 


S3 


0.003609 


89 


90 


0.171929 


X 


847 


-*- 


69804 


S3 


0.002086 


90 


91 


0.164525 


X 


462 


-*- 


69804 


S= 


0.001089 


91 


92 


0.157440 


X 


216 


■*■ 


69804 


S3 


0.000487 


92 


93 


0.150661 


X 


79 


-*- 


69804 


S3 


0.000171 


93 


94 


0.144173 


X 


21 


•*- 


69804 


3S 


0.000043 


94 


95 


0.137964 


X 


3 


•*• 


69801 


= 


0.000006 


1 95 



Total $13.235802 



Note.— The remarks that follow the table illustrating the calculation of the net single pre- 
mium that will insure $1 for life, apply to the value, at any age, of a series of annual payments 
of §1 for a designated term of years. 

Having shown, in the foregoing table, how we calculate the value, 
at age 50, of a whole-life series of annual premiums of $1 each, 
attention is called to the fact that the calculation of the net value at 



30 



NOTES 02s~ LIFE INSURANCE. 



any other age of a similar series of preniinms can be made, in a like 
manner, from any table of mortality, and at any rate of interest 

Value ot different Ages of a Life Series of Annual Payments of%\ 
each — American Experience Table of Mortality — Various Roto* 
of Interest. 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age, 


30 


'$19.5579 


$18.1726 


$16.9566 


$14.9325 


20 


21 


19.4520 


18.0665 


16.8862 


14 8846 


21 


22 


19.3420 


17.9968 


16.8126 


14.8342 


22 


23 


19.2277 


17.9032 


16.7355 


14.7811 


23 


24 


19.1089 


17.8055 


16.6548 


14.7252 


24 


25 


16.9854 


17.7036 


16.57d3 


14.6662 


25 


26 


18.8568 


17.5972 


16.4818 


14.6039 


26 


27 


18.7233 


17.4862 


16.3891 


14.5383 


27 


28 


18.5846 


H 3705 


16.2921 


14.4692 


28 


29 


18.4404 


17.2497 


16.1905 


14.3964 


29 


30 


18.2906 


17.1238 


16.0842 


14.3129 


30 


31 


IS. 1351 


16.9926 


15.9731 


14.2388 


31 


32 


17.9735 


16.8557 


15.8567 


14.1536 


32 


33 


17.8056 


16.7131 


15.7349 


14.0639 


33 


34 


17.6316 


16.5646 


vi.mrn 


13.9695 


34 


35 


17 4510 


16.4099 


15 .47 48 


13.8701 


35 


36 


17.2634 


16.2487 


15.3357 


13.7655 


36 


37 


17.0691 


16.0811 


15.1906 


13.6555 


37 


38 


16.8676 


15.9066 


15.0390 


13.5399 


38 


39 


16.6591 


15.7253 


14.8809 


13.4185 


39 


40 


16.4431 


15.5369 


14.7160 


13.2911 


40 


41 


16.2196 


15.3413 


14.5443 


13.1574 


41 


42 


15.9884 


15.1382 


14.3653 


13.0171 


42 


43 


15.7494 


14.9275 


14.17*0 


mstm 


43 


44 


15.5025 


14.7089 


13.9S49 


12.7160 


44 


45 


15.2477 


14.4826 


13.7834 


12.5549 


45 


46 


14.9849 


14.8481 


13.5741 


12.3865 


46 


47 


14.7144 


14.0065 


13.3573 


12.2109 


47 


48 


14.4362 


13.7569 


13.1327 


12.0279 


48 


49 


14.1507 


13.4998 


12.9008 


11.8327 


49 


50 


13.8583 


13.2358 


12.6617 


11.6404 


50 


51 


13.5595 


12.9651 


12.4159 


11.4365 


51 


52 


13.2546 


12.68S0 


12.1636 


11.2259 


52 


53 


12.9440 


12.4049 


11.9050 


11.0089 


53 


54 


12.6280 


12.1160 


11.6404 


10.7856 


54 


55 


12.3072 


11.8218 


11.3702 


10.5563 


55 


56 


11.9820 


11.5228 


11.0947 


10.3214 


g 


57 


11.6530 


11.2194 


10.8145 


10.0811 


58 


11.3207 


10.9121 


10.5299 


9 HzS 


58 


59 


10.9855 


10.6013 


10.2413 


9. 5858 


ffl 


60 


10.6481 


10.2877 


9.9493 


9.3317 


60 


61 


10.3092 


9 9718 


9.6544 


9.0788 


61 


62 


9.9695 


9 6544 


9.3574 


8.8127 


62 


63 


9.6296 


9 3360 


9.0587 


8 5490 

8.2801 


63 


64 


9.2901 


9.0172 


8.7590 


64 


65 


8.9518 


8.6^87 


B.4588 


8.0156 


65 


66 


8.6156 


8.3814 


8.1592 


7 7475 


66 


67 


8.2S22 


8.0662 


7.6607 


7.4792 


67 


68 


7.9524 


7.7536 


S 96a 


7.2:76 


68 


69 


7.6272 


7.4446 


7 2705 


6.9455 


69 


70 


7.3070 


7.1399 


6.9802 


6.6814 


70 



NOTES ON LIFE INSURANCE. 



31 



Value at different Ages of a Life Series of Annual Payments of%\ 
each — Actuaries' Table of Mortality — Various Mates of Interest. 



Age. 


Four per cent. 


Four and a half 
per ceut. 


Five per cent. 


Six per cent. 


Age. 


20 


$19.4504 


S18.0S86 


$16.8913 


$14.8943 


20 


21 


19.321)3 


17.9887 


16.8084 


14.8361 


21 


22 


19.2042 


17.8852 


16.7222 


14.7753 


22 


23 


19.0747 


17 7777 


16.6325 


14.7116 


23 


24 


18.9410 


1716663 


16.5391 


14.6451 


24 


25 


18.8027 


17.5508 


16.4422 


14.5756 


25 


26 


18.6593 


17.4311 


16.3416 


14.5028 


26 


27 


18.5122 


17.3069 


16.2364 


14.4267 


27 


28 


18.3597 


17.1783 


16.1274 


14.3472 


28 


29 


18.2022 


17.0450 


16.0141 


14.2641 


29 


30 


18.0397 


16.9070 


15.8963 


14.1772 


30 


31 


17.8718 


16.7640 


15.7739 


14.0866 


31 


32 


17.6985 


16.6159 


15.6469 


13.9918 


32 


33 


17.5196 


16.4626 


15.5149 


13.8928 


. 33 


34 


17.3350 


16.3039 


15.3778 


13.7894 


34 


35 


17.1443 


16.1393 


15.2354 


13.6812 


35 


36 


16.9476 


15.9689 


15.0880 


13.5681 


36 


37 


16.7443 


15.7923 


14.9333 


13.4497 


37 


38 


16.5342 


15.6092 


14.7730 


13.3258 


33 


39 


16.3172 


15.4193 


14.6064 


13.1961 


39 


40 


16.0929 


15.2224 


14.4329 


13.0601 


40 


41 


15.8610 


15.0181 


14.2523 


12.9176 


41 


42 


15.6212 


14.8060 


14.0641 


12.7681 


42 


43 


15.3736 


14.5862 


13.8684 


12.6116 


43 


44 


15.1186 


14.3591 


13.6656 


12.4484 


44 


45 


14.8571 


14.1255 


13.4562 


12.2789 


45^ 


46 


14.5896 


13.8857 


13.2408 


12.1035 


46 


47 


14.3170 


13.6407 


13.0200 


11.9227 


47 


4S 


14.0394 


13.3905 


12.7939 


11.7367 


48 


49 


13.7572 


13.1354 


12.5627 


11.5456 


49 


50 


13.4703 


12.8754 


12.3265 


113493 


50 


51 


13.1792 


12.6108 


12.0855 


11.1479 


51 


52 


12.8841 


12.3418 


11.8398 


10.9416 


52 


53 


12.5853 


12.0688 


11.5898 


10.7307 


53 


54 


12.2832 


11.7920 


11.3357 


10.5153 


54 


55 


11.9779 


11.5115 


11. 0*75 


10.2954 


55 


56 


11.6698 


11.2278 


10.8157 


10.0713 


56 


57 


11.3593 


10.9411 


10.5505 


9.8432 


57 


58 


11.0463 


10.6513 


10.2818 


9.6110 


58 


59 


10.7311 


10.3589 


10.0100 


9.3751 


59 


60 


10.4147 


10.0643 


9.7355 


9.1356 


60 


61 


10.0977 


9.7686 


9.4592 


8. 8935 


61 


62 


9.7805 


9.4721 


9.1815 


8.6492 


62 


63 


9.4641 


9.1755 


8.9033 


8.4033 


63 


64 


9.1489 


8.8794 


8.6249 


8.1561 


64 


65 


8.8356 


8.5845 


8.3520 


7.9083 


65 


66 


8.5248 


8.2913 


8.0700 


7.6605 


66 


67 


8.2170 


8.0003 


7.1946 


7.4131 


67 


68 


7.9130 


7.7123 


7.5215 


7.1668 


68 


69 


7.6130 


7.4276 


7.2509 


6.9219 


69 


70 


7.3172 


7.1462 


6.9831 


6.6785 


70 



To obtain the net Annual Premium. — "When the net single pre- 
mium that will insure $1 for whole life has been calculated, and the 
value at the designated age of a whole-life series of annual pay- 
ments of $1 each is known, we obtain the net annual premium that 
will insure $1 for whole life at that age by the following rule: 
Divide the net single premium at any age by the value at that age 
of a whole-life series of annual payments of $1 each, and the result 
is the net annual premium that will insure $1 for whole life at that 
age. For instance (American Experience, A\ 2), at age 30, the net 



- 



NOTES ON LIFE INSURANCE. 



premium to insure $1 is $0.262609, the value at age 30 of the 
series of $1 premiums is $17.1238, therefore we hare the proportion : 
$17.1238 : $0.262609 ::$1 is to the net annual premium that at age 
30 will insure $1 for whole life. 



:~:tz — I: 




The following table shows the net annual premiums that will 
insure $1000 at different ages, from 20 to 70 inclusive : 



Ifot Annual Premiums — American Experience — Va 

1 '■:*-;■*'.*:. 



■i B-jU 



^ -—^ 




J:-;-:-:. 


S— ;;: : :~: 


i:: 


.?<-, {• 2 VS 


<:: »; 


fill : : r' 


$:: 'sa 


20 


*1 - ~^~ 


a » 


11. 601 


:: :.s; 


a 


22 : -:- 


12 aa 


:: H 


:: s* 


22 


23 '■': T A~ 


12 ~i4 


:•: :i4 


:: :-:•: 


23 


24 i: -": 


:s ::•: 


:c 424 


:: ?i:t 


24 


a --- '-'-'- 


:• 4.22 


:: 3: 


:: :*:• 


23 


2* a :-: 


1: 735 


I": X-i 


11 Kl 


26 


•;- a Az 


14 12: 


a i:rT 


:: :s: 


27 


2S : z A~ 


14 -.:- 


a M 


:: :•:>= 


a 


29 1: ":T 


:4 ri: 


14 14: 


1- *■•:<? 


29 


m : 21: 


E B 


14 :;:4 




30 


>: :■: zs: 


1: 7*7 


14 rN: 


' : -'■;'"■ 


31 


s : - :"5 


H BE 


1-: 44! 


14 ir.: 


33 


ST 


1: 71 


1: r:24 


14 =-:•: 


i-: 


:4 :« £.; 


:- n:* 


I: 4-:5 


14 ?:- 


34 


Si' '-' '-a- 


:: «-? 


:: :••: 


1: 4^4 


i: 


95 :.- 4*4 


:» *»t 


:: =.*s 


1: "42 


a 


r 2: :.4 


:.- 12.: 


:? ■-:: 


1: B9 


c- 


35 -' S-4 


:- s:e 


:- »?" 


17.2a 


c> 


» :: v.-: 


2: :.2r 


lr :.»: 


:: :-v: 




4-: 22 i-:4 


a M 


•:■: :;-4 


1- :.ic 


4; 


41 £ IrC 


2: 121 


•:: -yy 


::- :=:?> 


41 


4: -4 :n4 


2: Hi 


'- ". r'.-C 


•i'.: ;:« 


at 


4l 24 rW« 


21 ?2? 


2: : '"> 


•:: :•>• 


43 


44 2* i44 


•14 >;4 


::: ss: 


a B1 


-4 


4^ •:- :i2 


E Hi 


14 r£J 


•J: :^: 


4 r 


4: is ■:--: 


2: ::: 


Z': ;r.; 


24 1-r 


46 


4 _ :•.:- 4r'r 




• v- ^J." 


2.: i:: 


a 


4- y: - •- 


■-•; - ■-■: 


C> " 2T 


a a 


4* 


49 lOT 


>i ::-3 


•":- »: : ' 


2- v': 


49 


V £ :-> 


::= 4A 


s: i'; 


a « 


'<:• 


51 ■ :■' -S« 


>4 Vi* 


a as 


y: Si: 


B 


' ' '.- - ■'- 


S EB 


:4 :.S 


3i 4"t 


M 


\l y Hi 


r v: 


a aa 


>4 aa 


CO 


;4 a! "2? 


>:.- 4-E 


is w 


a na 


f . 


" 4 1 r"2 


4: :.:■: 


4! -ii! 


» 12^ 


55 


:■: 4r^ r'r" 


41 B9 


4Z : 14 


4: 2-2 


a 


:" 4" :■'•- 


4: >::.i 


44 S4r 


42 :..>: 


a 


'■* 4: -'"- 


4- rr* 


^~ : ^.- 


4- as 




r . : :,2 :o» 


a aa 


v :'fs 


4: ::: 


; : 


"V :•: 4: 2 


:4 141 


li f..-: 


.:.' :.!.» 




CI | :•- r .;'i' 


r 22: 


c. r *.j 


cc BB 




" '- *4 4; 


-:•: : ? 


!.r -:4-5 


Cr KB 


H 


-£ :. r i.v 


:4 :*: 


a ra 


60.30 


•it 


A :■- 1* 


r si> 


:>! :4r 


C4 124 


A 


:. r -': 24- 


r. *:-* 


-: :-:o 


r'S 1:2 


& 


■y ~.~ y.~ 


": U4r 


-4 .-4; 




cc 


ir 2"- 


s: >:s 


" : yy-- 


- ::•:• 


r. 


:■: \- 2S: 


s: .-:: 


^ :cS 


-2 >.2 


0; 


: : : ?: z ~-i 


r. •*:= 


v- >:4 


»T £" T 


■ 




>: .-'.^ 


E tfi 


S >;c 


:: 



NOTES ON LIFE INSURANCE. 



33 



Net Annual Premiums — Actuaries' — Various Hates of Interest, 


Age. 


Pour per cent. 


Four and a half 
per cent. 


Five per cent. 


Six per cent. 


Age. 


20 


$12,948 


$12,221 


$11,583 


$10,536 


20 


x>i 


13.273 


12.528 


11.875 


10.799 


21 


22 


13 610 


12.850 


12.182 


11.077 


22 


23 


13.963 


13.188 


12.502 


11.370 


23 


24 


14.8:34 


13.543 


12.843 


11.678 


24 


25 


14.722 


13.915 


13.200 


12.004 


25 


26 


15.129 


14.307 


13 576 


12.349 


26 


27 


15.557 


14.718 


13.971 


12.712 


27 


28 


16.005 


15.151 


14.387 


13.096 


28 


29 


16.477 


15.606 


14.826 


13.502 


29 


80 


16.972 


16.085 


15.289 


13.932 


30 


31 


17.492 


16.589 


15.776 


14.386 


31 


32 


18.040 


17.121 


16.291 


14.867 


32 


33 


18.616 


17.681 


16.835 


15.376 


33 


34 


19.225 


18.273 


17.409 


15.916 


34 


35 


19.866 


18.898 


18.018 


16.490 


35 


36 


20.544 


19.559 


18.662 


17.099 


36 


37 


21.260 


20.260 


19.345 


17.747 


37 


38 


22.018 


21.003 


20.072 


18.439 


38 


39 


22.823 


21.791 


20.844 


19.176 


39 


40 


23.677 


22.630 


21.667 


19.965 


40 


41 


24.586 


23.524 


22.545 


20.810 


41 


42 


25.554 


24.478 


23.484 


21.716 


42 


43 


26.585 


25.490 


24.487 


22.683 


43 


44 


27.682 


26.580 


25.558 


23.728 


44 


45 


28.845 


27.732 


26.696 


24.837 


45 


46 


30 080 


28.954 


27.905 


26.017 


46 


47 


31.385 


30.248 


29.186 


27.270 


47 ' 


48 


32.767 


31.618 


30.543 


28.597 


48 


49 


34.227 


33.068 


31.982 


30.009 


49 


50 


35.775 


34.605 


33.507 


31.508 


50 


51 


37.415 


36.235 


35.124 


33.099 


51 


52 


39.151 


37.963 


36.842 


34.790 


52 


53 


40.996 


39.796 


38.664 


36.586 


53 


54 


42.950 


41.741 


40.598 


38.495 


54 


55 


45.025 


43.807 


42.654 


40.527 


55 


56 


47.230 


46.003 


44.839 


42.688 


56 


57 


49.571 


48.337 


47.163 


44.989 


57 


58 


52.067 


50.823 


49.640 


47.444 


58 


59 


54.724 


53.473 


52.281 


50.062 


59 


60 


57.556 


56.299 


55.098 


52.858 


60 


61 


60.572 


59.307 


58 098 


55.838 


61 


62 


63.7.82 


62.511 


61.294 


59.014 


62 


63 


67.199 


65.923 


64.698 


62.397 


63 


64 


70.841 


69.558 


68.324 


66.004 


64 


65 


74.718 


73.427 


72.186 


69.845 


65 


66 


78.846 


77.546 


76.297 


73.936 


66 


67 


83.237 


81.933 


80.675 


78.293 


67 


68 


87.913 


86.601 


85.333 


82.928 


68 


69 


92.892 


91.572 90.295 


87.866 


69 


70 


98.202 


96.87S 


95.585 


93.131 


70 



34: NOTES ON LIFE INSURANCE. 



CHAPTER n. 

COMMUTATION TABLES. 
Method by which the Values placed ra the Columns headed 

RESPECTIVELY C. M. R. D. X. AND S. ARE COMPUTED. 

Nearly one hundred years ago, it was noticed that, by commenc- 
ing the calculations at the oldest age given in the tables of mortali- 
ty, and then taking an age one year younger, and so on decreasing 
the age successively one year to the youngest age. the construction 
of the commutation-tables would "be greatly facilitated, provided the 
numerator and the denominator of the fraction which gives the 
amount that will, at each age, insure $1, is multiplied by a quantity 
obtained by raising the amount that will produce $1 in one year to 
a power, the exponent of which is the age for which the calculation 
is being made. For instance, the greatest age given in the Ameri- 
can Experience Table of Mortality is 95. Interest being 44 per cent 
per annum, the amount that will at age 95 insure $1 for one year is 
equal to y.^^- multiplied by the fraction which at age 95 represents 
the chance that the insured will die during the first year. By this 
table, the whole number living at age 95 is 3, and the number of 
deaths during the year is 3 ; therefore, we may express the amount 

that will, at age 95, insure $1 for life by the fraction T,¥4 ; • 
Multiply the numeratoi and denominator by (y.-fe-) 95 , which will not 

change the value of the fraction, and it becomes : - : ^ 5 — ill . This 

<T.ArrX3 

is the expression used to designate the amount that will at age 

95 insure $1 for whole life. 

The value of (y.oVs) 86 * 5 shown in the appended table opposite 96 

years. This value is -$0.014616 ; multiply this by 3, which is the 

number of deaths between age 95 and age 96, and we have for the 

numerator of the above expression 10.043549. This is called C 85 , 

and is placed in the commutation-table in the column headed C. and 

opposite to age 95. (See pages 180, 182.) It is also placed in that 

column of the same table which is headed M, and opposite age 95. 

The denominator of the above expression is obtained by taking the 



NOTES ON LIFE INSURANCE. 35 

value of (t.tsVt) 98 , which is $0.015274, and multiplying it by the 
number of persons living at age 95, which is 3. The result of this 
multiplication gives $0.045822. This is called D 95 , and is placed in 
that column of the table which is headed D, and opposite to age 95. 
Next, take an age one year younger — namely, age 94. The 
amount that will, at age 94, insure $1 for the first year is expressed 
by y.otg- multiplied by the fraction which at age 94 represents the 
chance that the insured will die before he is 95. From the table, we 
find that the number living at age 94 is 21 ; of this number, the table 
shows that 18 will die before age 95 ; therefore, the fraction, which, 
at age 94, represents the chance that the insured will die during the 
first year, is Jf ; from which we see that at age 94 the amount that 
will insure 8l for the first year is expressed by T-oVgrX — t ^ 1Q 

amount that will at age 94 insure $1 for the second year is equal to 
(t-oVf) 3 multiplied by the fraction which at age 94 represents the 
chance that the insured will die between age 95 and age 96. By 
the table, the number of deaths during this year is 3; the .number 
living at age 94 being 21, it follows that the fraction which, at age 
94, represents the chance that the insured will die between age 95 

and age 96 is g 3 T . Therefore, the fraction vr.irrg) X ex p resses t he 

2 1 
amount that will, if paid at age 94, insure $1 for the second year. 
This carries us to the limit of the mortality table. Therefore, 

±±£2 J~lT-or57 a- expresses the amount that will, if paid in 

hand at age 94, insure $1 for whole life. Multiply the numerator 
and denominator of this fraction by (t.oVs") 9 \ anc l it becomes 
(T.o 1 4T) 9 'Xlg+(T--oVT) 96 X3 _ The second term f the num erator at 

(t.oWXSI 
age 94 is identical with the numerator previously found at age 95 ; 
therefore we have only to calculate the first term of the numerator 
at age 94 and add it to the numerator at age 95, in order to obtain 
the whole numerator at age 94. 

In order to obtain the first term of the numerator at age 94, we 
multiply the decimal value already found for (y.^ 1 ^) 95 by IS. This 
is called C 94 . It is placed in the column headed C and opposite to 
age 94. Add this to the numerator at age 95, call the result M 94 , 
and place it in the table opposite age 94 in the column headed M. 
The denominator at age 94 is obtained by multiplying the decimal 
value of (y.oV r) 94 ^7 21 > tne number living at age 94. The result is 
$0.335188. This is called D 94 , and it is placed in the column headed 
D, opposite age 94. 



36 NOTES ON LIFE INSURANCE. 

M g4 divided by D 94 is the net single premium that will at age 94 
insure $1 for life. 

The expression which gives the amount that will at age 93 insure 

$1 for the first year is iJ *" 5 , X 58 . The amount that will, if paid at 

age 93, insure $1 for the second year is expressed by vr-04sJ X 

The amount that will, if paid at age 93, insure $1 for the third year 

r i y X 3 

is expressed by Vl, ° 45 ^ . We have reached the table limit ; 

hence, adding together the above respective yearly amounts, we find 
that T-oVir X 58 + (y.Ar)' X 18 + ( T .p^) 3 X_3 expresges the 

amount that will, if paid in hand at age 93, insure $1 for whole life. 
Multiply the numerator and denominator of this fraction by ( T . 0V5-) 93 , 
and it becomes : 

(y.fe)" X 58 + ( T ,fe)" X 18 + (t,^)- X 8 . 

(l.Al)" X 79 

The second and third terms of the numerator of this fraction are 
together equal to the numerator at age 94 ; therefore, we have only to 
calculate the value of the first term of the numerator at age 93, and 
add it to the numerator at age 94, in order to obtain the numerator 
at age 93. In making this calculation, take the decimal value of 
(t.04t) 94 from tne table, ifc is $0.015961 ; multiply it by 58; call the 
result C 93 , and place it in the column headed C, and opposite age 93. 
Add C 93 to C 94 and C 95 , and call the sum M 93 , and place it in the 
table opposite age 93 in the column headed M. To calculate the 
value of the denominator at this age, take the decimal value of 
(t-oVt) 93 fr° m the table and multiply it by 79 ; call the result D 93 , 
and place it in the table opposite age 93 in the column headed D. 

In like manner, at each successive younger age, calculate the nu- 
merator and denominator, multiply both by y.-^g- raised to a power, 
the exponent of which is the age for which the calculation is being 
made, and it will be found at each age that it will only be necessary 
to calculate the first term of the numerator of the general expres- 
sion, and add to it the numerator of an age one year greater, in order 
to obtain the numerator at the age in question. The denominator at 
each age is calculated by raising T .^±j to a power, the exponent of 
which is the age, and multiplying the result by the number living at 
that age. The values of M and D at each age having been calculat- 
ed, to obtain the net single premium at any age, we have only to di- 
vide M at that age by D at the same age. 



NOTES ON LIFE INSURANCE. 37 

The N Column. — It will be remembered that, in order to convert 
the net single premium that will insure $1 for life into an equivalent 
annual premium, it was stated to be convenient first to obtain at each 
age the value in hand of a life series of annual payments of $1 each, 
the first being paid in advance, and $1 paid at the beginning of each 
following year, as long as the person lives to make the payment. 
The method of calculating the commutation-tables for determining 
the value at each age of such a life series of annual payments will 
now be explained. 

We again assume that the table of mortality used is the American 
Experience, and that the rate of interest is 4£ per cent per annum. 
The first payment is $1 in hand. At age 95, there are by the table 
3 persons living. The value of the first payment is $1. This may 
be expressed by f . Multiplying both numerator and denominator of 

this fraction by (y.-Jo-) 95 , tne expression becomes vS JgiL_X =$1. 

W.oV5) J& X3 

There can in this case be no second payment, since, by the table, all 
living at age 95 die before they reach 96. The denominator of the 
preceding fraction is identical with the denominator previously called 
D 95 . The numerator of the same fraction is also identical with D 96 . 
This numerator is called N 95 , and is placed in the table in the column 
headed N, and opposite age 95. In this case, no calculations are re- 
quired to be made, because D 9a has been previously determined, and 

we have =3* = ^ = $1. 

To find the value at age 94 of a life series of annual payments of 
$1 each. — The number living at age 94 is 21 ; the value of the first pay- 
ment may be represented by -fy. The value at age 94 of a payment 
of $1 to be made at age 95, in case the person is alive to make the 
payment, is expressed by y.-^-g- multiplied by the fraction which at 
age 94 represents the chance that the person will be alive at age 95. 
The table shows that, of 21 persons alive at age 94, 3 will be alive 
at age 95 ; therefore, the value at age 94 of the payment of $1 to be 

— — — V 3 
made at age 95, is expressed by liML-Ll — . Add this to the value of 

the first payment as expressed above, and we have the value of the 

life series at age 94 expressed by vt.qtt) X — Multiply both 

it j. 

the numerator and denominator of this fraction by ( T .-fe-) 9 *, and we have 
(T-fcr) 94 X 21 +( 1T J4 T ) <J5 X 3. Thig , h j 94 f ]ife 

(Tin,) 94 X 21 
series of annual payments of $1 each. The denominator is identical 



38 NOTES ON LIFE INSURANCE. 

with the quantity we have previously called D 94 , and which has been 
calculated and placed in the D column opposite age 94. Call 1ST 94 the 
numerator at age 94. It is seen that the first term of the numera- 
tor of the above expression is identical with the denominator. The 
second term of the numerator is identical with D 05 ; therefore, the 

expression may be written : -——= — ^— *.». The value of D aA and 

r j D D 94 

^9 4 -"^94 

of D 95 being already known, we have in this case no other calcula- 
tion to make than to add together the two terms of the numerator 
and place the sum opposite age 94 in the column headed ~N. 

In like manner, at age 93, to obtain the value of a life series of an- 
nual payments of $ I each, represent the first payment by the number 
living at age 93, divided by the number living at the same age ; 
represent the second by T ^ F multiplied by the fraction which at 
age 93 expresses the chance that the second payment will be made ; 
the third by vr.-fe) 2 multiplied by the fraction which at age 93 ex- 
presses the chance that the third payment will be made. Add the 
three yearly values together. Their sum is the value in hand at age 
93 of a life series of annual payments of $1. Multiply both nume- 
rator and denominator of the fraction which expresses this value by 
(t-tit) 93 an( ^- we °^^ a i n an expression the denominator of which is 
identical with D 93 previously calculated. We find, too, that the 
first term of the numerator is identical with D 93 , that the second 
term of the numerator is D 94 , the third term D 95 . Therefore, at age 

93, calling the numerator N 93 , we have • 93 = - 93 — — 94 — ^ 5 



I> 93 ^03 

In like manner, at age 92, we find:- 9 ?— 92 + 93 _^ 94+ — **. ; and 

-^92 -^92 

so on for each successive younger age to the table limit, where we 

fca- y » p ,.+ p ..+p.,+***+p»+p„ 

The R Column. — In the table there is a column headed R. This 
is formed by adding to M at any age the sum of the Ms at all older 
ages. R at any age divided by D at the same age is the net single 
premium that will insure $1 the first year, $2 the second year, $3 the 
third year, and so on, increasing the insurance $1 each year to the 
table limit. 

The S Column. — In the table there is a column headed S. This 
is formed by adding to N at any age the sum of the Ns at all older 
ages. S, at any age, divided by D at the same age, is equivalent in 
value to a life series of annual payments of $1 in hand, $2 at the 



NOTES ON LIFE INSURANCE. 39 

beginning of the second year, $3 at the beginning of the third year, 
and so on increasing the payments by one dollar eacli year to the 
table limit. 

Note.— It is very important that the foregoing method of constructing the C. M. R. D. N. 
and S. columns be well understood. No person can comprehend the formulas and rules now 
generally used in making calculations of life insurance net values without first forming defi- 
nite ideas in regard to the real meaning of these columns, and the manner of computing the 
quantities therein represented. 

In illustration of the manner of using tlieCommntation Columns. — 
Suppose the age of the insured is 30 at the time he pays his net 
single premium, and that the insurance is not to commence until he 
is 40, and is then to continue for life : the amount that will at age 30 
insure $1, to be paid to the heirs of the insured in 11 years, provided 
he dies between age 40 and age 41, is expressed by (-J-.-0V5-) 11 multi- 
plied by the fraction which at age 30 represents the chance that the 
insured will die between age 40 and age 41. The amount that will at 
age 30 insure $1 between age 41 and age 42 is expressed by ( T .-oYg ) ia 
multiplied by the fraction which at age 30 expresses the chance that 
the insured will die between age 41 and age 42. In like manner, the 
amount is expressed for each year to the table limit. If we multi- 
ply the numerator and denominator by ( t .ttf) 3 ° tne numerator be- 
comes equal to M 40 and the denominator to D 30 . Therefore, the 
amount that will at age 30 purchase an insurance of $1, beginning 
at age 40 and' continuing from that time to the table limit, is ex- 

pressed by =~. 

■^30 

The net single premium that will at age 30 insure $1 for whole 

M 

life is expressed by— *±. Therefore, the net single premium that 

30 

will at age 30 insure $1 for 10 years is 30 4 °. 

If the insurance is effected at age 30, to begin at age 40, and con- 
tinue until age 70 ; the net single premium in this case will be ex- 
pressed by 1 40 ™ 

30 

The value at age 30 of a series of annual payments of $1 each for 

]sr isr 

20 years, provided the person lives so long, is expressed by — ^ — -• 

N" 
Because =— 3 ^ is the value at age 30 of the whole-life series, and 

30 

jy- 5 - is the value at age 30 of that portion of the series that is be- 

30 

yond the 20 years. 



40 NOTES OX LIFE INSURANCE. 

If the annual payments beginning at age 50 are only to continue 
until age 70, the expression becomes ^ "L* "*" 7 . 

The net annual premium that will at any age insure %\ for a 
designated term of years is obtained by dividing the difference be- 
tween M at the beginning and M at the end of the term, by the dif- 
ference between X at the beginning and N" at the end of the term. 
For instance, to insure at age 30, $1 for twenty years. The amount 

x so — *• in hand at age 30 is the equivalent of a series of annual 

payments of 81 for twenty years. The net single premium is, 

Therefore we have the proportion, 



M — M, 



D. 



N„. M;„— M„ :il M.— M, 



I> L> N — N e 



Endowment combined with Term Insurance. — Suppose the age is 
30, and the endowment is payable, if the insured is alive, at age 60 : 
the amount that will at 4|- per cent produce Si in thirty years is ex- 
pressed by (t.itj-s) 30 . The fraction which at age 30 expresses the 
chance that the insured will be alive at age 60 is the number living 
as shown by the mortality table at age 60, divided by the number 
living at age 30. By the American Experience table, this fraction is 

5<7917 . Therefore (t.JjtY° X 57917 . g ^ net sni „i e pre mium 
85441 85441 

that will effect the endowment. Multiply both numerator and de- 
nominator of this fraction by ( j.-^j ) 30 and we have, 

^ T - 045 ^ — _ — ' = . «°. A similar expression is obtained in case 

(T.fe) 3 °X 85441 D 30 P 

the endowment is effected at any age and is payable in any given 
number of years. 

Therefore to obtain the net single premium at any age for an en- 
dowment payable at any greater age, divide D at the age when the 
endowment is payable by D at the age when the insurance is effected. 
For instance, at age 20, the net single premium that will, if paid at 
that time, insure an endowment of $1 at age 45 is expressed by 

_ I- 5 * The net single premium that will at age 20 insure $1 for 

25 years is expressed by 20 ^ 45 - Add this to the net single 
premium that will effect the above endowment, and we have 
i?« _|_ ^ao " 1 4j. "j^g i s an expression for the amount that will, 



NOTES ON LIFE INSURANCE. 41 

if paid at age 20, insure $1 to be paid to the heirs of the insured, at 
the end of any year in which he may die, provided he dies within 
25 years, and insure $1 to be paid to himself if alive at the end 
of the 25 years. 

The net annual premium that will effect this endowment and term 
insurance is obtained from the proportion : 

N,-K )t . D„ + M M -M.. .. g . P 4 » + M M -M 4t 

»» D.. N,.-N„ ' 

The net single premium that will at age 30 insure $1 the first 

year, $2 the second year, $3 the third year, and so on, increasing the 
insurance $1 each successive year for life, is expressed by — l° • The 

net single premium that will at age 30 insure $1, beginning at age 
40, $2 at age 41, $3 at age 42, and so on, increasing the insurance $1 

each successive year for life, is expressed by — !• . 

^ 30 

N Therefore — -22. —^ is the amount that will at a%e 30 insure $1 

d m d so 

the first year, $2 the second, $3 the third, and so on until age 40, and 
continue to insure $10 for life. 

The net single premium that will at age 30 insure $10 to begin at 

io yc M 

age 40, and then continue for life, is expressed by — — *£ • There- 

-^30 

fore — a° — — i» * — i? is the net single premium that will at 

"^30 "^30 ^30 

age 30 insure $1 the first year, $2 the second, $3 the third, and so 
on, increasing the insurance $1 each year for ten years ; the insur- 
ance to cease at the end of that time. 

For further illustration of the use of the commutation columns, 
see algebraic discussion and formulas. 



42 NOTES ON LIFE INSURANCE. 



CHAPTER III. 

TRUST FUND DEPOSIT, OR "RESERVE," AS IT IS USUALLY CALLED. 

This fund has by high authority been well styled "the great 
sheet-anchor of life insurance." By referring to the table of net 
single premiums (page 25), it will be seen that by the American 
Experience Table of Mortality, and 4| per cent interest, the net 
single premium that will insure $1000 for whole life, at age 20, 
is $217,448. At age 21, the net single premium is $221,155. 
The latter sum is the net amount that must be charged by the com- 
pany in order to insure a person who is 21 years old. This is the 
sum that the company must hold at the end of the first year, upon 
the policy issued at 20, after paying the net cost of insurance for 
the year. The net single premium $217,448 paid at age 20 will, 
when increased by net interest for one year, furnish the required 
contribution of this policy to pay death claims, and leave in the 
hands of the company the net single premium necessary to effect 
the insurance for whole life at age 21, in case death did not occur 
before. 

At the end of the second policy year, when the policy-holder will 
be 22 years old, the net single premium that will then insure $1000 
for whole life is $225,019, and this is the amount that must then be 
held by the company to the credit of this policy, if the insured is 
still alive. In like manner, each successive year, if the policy-holder 
survives, the company must have, in order to comply with its con- 
tract, an amount on hand to the credit of this policy equal to the 
net single premium that will at the age the policy-holder has at- 
tained be sufficient to effect the insurance. 

Deposit or " Reserve" in case a Policy is paid for by equal Annual 
Premiums. — From the table (page 33) it is seen that at age 42 the net 
annual premium that will insure $1000 for whole life (Actuaries' 4 
per cent) is $25,554. This premium is to be paid at the beginning of 
each year, as long as the person is alive to make the payment. At 
the end of the first year, or beginning of the second — supposing the 
insured to be alive — he pays the net annual premium, $25,554, and 
is insured for another year; but he is now 43 years old, and $26,585 
is the net annual premium required to insure $1000 for life at age 43. 



NOTES ON LIFE INSURANCE. 43 

Why is it that the man who was insured at age 42, and who lias 
been insured one year, and lias paid for that insurance, can, at 
43 years of age, be insured by the company for a less premium than 
is required to insure a man of the same age, 43, but who now takes 
out a policy for the first time in that company ? Taking, for further 
illustration, a still greater age, we find that at age C5 the net annual 
premium that will insure $1000 for life is $74.718 ; and yet the per- 
son who took out his policy at age 42, supposing he is still alive, 
can be safely insured at age C5 by the company for a net annual 
premium of $25,554. 

How is this ? Why is it that a man 65 years of age can be in- 
sured safely by a company for a net annual premium of $25,554, 
and another man of the same age, probably in better health, be- 
cause he has just passed a medical examination, can not be safely 
insured by the company for a less net annual premium than $74,718 ? 
The net annual premium is calculated to provide against all the 
jn-obabilities and risks of the insured dying in any year, and of his 
policy becoming due ; and also the risk of his being alive, from year 
to year, to pay his annual premium. At the end of each year, after 
the net annual premium has paid its proportion of the losses by death 
for the year, there must be in the hands of the company, on account 
of and to the credit of each and every outstanding policy, an 
amount in money or securely invested funds, that will be in present 
value the equivalent of a life series of annual premiums, each of 
which is equal to the difference between the net annual premium the 
insured paid on taking out his policy and the net annual premium he 
would now have to pay if he were taking out a new policy at his 
present advanced age. This amount that must be in the hands of 
the company at the end of each year's business, to the credit of the 
respective policies, is variously styled, by life insurance writers, " re- 
serve," "reserve for reinsurance," "net premium reserve," "net 
value," " true value," " self-insurance," etc., etc. 

As before stated, the net annual premium to insure $1000 at age 
42 is $25,554, and the net annual premium to insure the same 
amount at age 43 is $26,585. The difference between these two 
premiums is $1,031. The value at age 43 of a life series of annual 
payments of $1 is from the table (page 31) found to be $15,374. We 
can find the value at the same age of a whole-life series of annual 
payments, each of which is equal to $1,031, by the proportion: $1 : 
$1,031 : : $15,374 is to the answer. Solving this proportion, we find 
that the value at age 43, of a life series of annual premiums of $1,031, 
is equal to $15,851. 



rs 


of 


age, is 


obtained by 


en 


oe : 


is $2.12 


S, and there 




at 


at the 


end of the 


^t 


tn 


at bme 


of a whole- 






us e q n 


al to $2.12 S, 


Cl 


pri 


emtum. 


due to age 


Y: 




i the ta 


.ble, we find 


'■ 01 




rt annu 


al premiums 


Sa 


me 


asre of 


a whole-life 


s 


>.l: 


-:i:y 


the propor- 



14: NOTES OS LIFE INSURANCE. 

If the company has the $15,851 on hand in deposit, which is the 
cash equivalent of this difference in the future net annual premiums; 
this amount of cash in hand, together with the smaller net annual 
premium due to the age 42, is just the same value as the net annual 
premium due to age 43. This $15,851 is the amount that must be 
held on deposit in trust for the policy of $1000 taken out at age 42, 
at the end of the first year of the policy. 

The net annual premium at age 44 to insure $1000 for whole life 
is found from the same table to be $27,682. The difference between 
the net annual premium due to age 44, and that which the person 
insured at age 42 will pay vrhen he is 44 years of age, is 
subtracting $25,554 from $27.682 ; this diffe] 
must be in the hands of the company a de 
second year of this policy equal to the value 
life series of annual premiums, each of u 
which is the difference between the net anm 
44 and that which the insured will pay. 
that the value at age 44 of a whole-life eerie 
of $1 each is $15,119. We find the value a: 
series of annual premiums, each of which is 
tion : $1 : $2,12 5 : : $15,119 is to the answer. 

Solving this proportion, vre find the value sought is $32,172. 
This is the fund on deposit, or the '*' reserve," for this policy at the 
end of the second year. In a manner entirely similar, we find the 
amount that must be on deposit at the end of each policy year ; and 
if the company has it on hand, and keeps it securely invested at the 
net rate of interest, and regularly compounds the interest yearly, 
this "trust fund deposit." together with the present value of the fu- 
ture net annual premiums, will always keep the policy that is pay- 
ing the smaller net annual premiums due to the younger age at 
which the holder entered the company, just on a par with those poli- 
cies that come in later, or at a more advanced age of entry, and pay 
the larger annual premium due to this advanced age. 

The amount of this deposit may be calculated by a somewhat dif- 
ferent process, as follows: At the time the contract is entered into. 
value at that time of the whole-life series of net annual premiums 
to be paid for the policy is exact 1 to the net single premium 

at that age. Remember that the net single premium is obtained by 
direct calculation for insurance each separate year, and we convert 
the net single premium into an equivalent net annual premium. At 
the end of any policy year, find the net single premium due to that 
age, then find the value at that age of the series of net annual pre- 



NOTES ON LIFE INSUKANCE. 45 

miums the insured is to pay ; this will be less than the net single 
premium that at that age will effect the insurance, and this difference 
is the amount that must be held by the company in deposit to the 
credit of the policy. For instance, at age 42, the net single premium 
to insure $1000 for whole-life, actuaries' 4 per cent, is $399,184, and 
this is the value at that ago of the whole-life series of net annual 
premiums, $25,554, due to the age. At the end of the twentieth 
policy year, the insured, if alive, will be 62 years old. The net sin- 
gle premium required to insure $1000 for whole-life at 62 is $623,826. 
Now the value at age 62 of a life series of annual payments of $1 each 
is $9,781 ; multiply this by the net annual premium the insured is pay- 
ing, that is, $25,554, and we have the value at age 62 of the series of 
net annual premiums the insured is to pay. This amounts to 
$249.931 ; but direct calculation shows, as stated above, that the in- 
surance on the assumed table of mortality and rate of interest can 
not be effected at that age for less than $623,826 net single premium. 
The difference between this sum and the value of the series at age 62 
which the insured is to pay must be held by the company in deposit 
to the -credit of the policy; therefore, subtracting $249,931 from 
$623,826, we have $373,895, which is the sum that must be in deposit 
at the end of the twentieth policy year belonging to a policy taken 
out at age 42 for $1000. 

It is necessary to have the value of the net single premium at each 
age, and all that portion of this value not in the present value of the 
future net annual premiums must be on hand in deposit. Therefore, 
a company that charges a less net annual premium than that called 
for by the table of mortality and rate of interest designated by law, 
must be required to add to what would be the legal deposit, in case 
the future net premiums are equal to those required by the law, an 
amount equal to the value at that time of a series of annual premiums 
each of which is equal to the difference between the net annual pre- 
mium called for by the legal data and the net annual premium the 
company has agreed to receive. 

Illustration. — To illustrate the manner in which the "deposit" 
must accumulate in the earlier years of a life insurance company, in 
order to enable it to meet its obligations when the death-claims ex- 
ceed the premiums, let us suppose that a company insures twenty 
thousand policy-holders for five thousand dollars each, at age thirty. 
The net annual premium required for each person is $84.85. This, 
on 20,000 policies, would make the first payment of annual premiums 
amount to $1,697,000. The net interest is assumed to be four per 
cent, and, for the first year, it amounts to $67,880. The company 



NOTES OX LIFE EN'SUE-AyCE. 

L:i5. r/.fi-rfn-r. :.: :::r enil :f the £rst year. Sl.7o4.SS-j. By the table 
of mortality, 168 of the insured will die daring the first year ; to the 
heirs of each, the company must pay five thousand dollars. The 
losses by death are, therefore, $840,000 ; leaving on hand with the 
company, after all the death-claims are paid, $924,880 ; which would 
be a handsome "surplus" at the end of the first year's business, but 
for the fact that every dollar of this sum belongs to the trust fond 
deposit, and is an already accrued liability — a debt. 

At the end of the thirty-fourth year, the deposit for each outstand- 
ing policy must be $2464.25. The table of mortality shows that 
1 . . - 1 : the policy-holders will be living at the end of the thirty- 
fourth year ; the company must, therefore, have on hand a trust fund 
deposit amounting to $27,838,6oi.i ". We find thai 11,743 policy- 
holders were living at the beginning of the thirty-fourth year, and 
their net annual premiums amounted, in the aggregate, to $996,308.70. 
There were 445 deaths during the year, and the aggregate losses by 
death amount e 3 . : % . . . _ 5 . 3 00. Thus, in this year, the death-claims 
exceed the annual premiums by more than one and one quarter mil- 
lions of dollars. But the company has on hand, in deposit, .at the 
end of the yean $27,838,632.25, : : r r having paid the death-claims. 
The company, however, is not rich, nor more than able to pay its 
liabilities, because it will surely take the last cent of this amount, 
with all the fature net annual premiums, and interest compounded 
regularly all the time, to enable it to meet and pay its now rapidly 
increasing death-claims. 

Let us look into the accounts of the company at the end of the fif- 
tieth year. The u deposit" on account of each policy at the end of 
this year is $3708.20 ; and there are living 3080 policy-holders. The 
aggregate "deposit" for the outstanding policies at this time is 
$11,421,256. Their were 4*51 deaths during the year, and the aggre- 
gate of poL::ir- :"_;.: matured during the year amounted to $2,305,000. 
There were 3541 policy-holders living at the beginning of the year, 
and the aggregate of the net annual premiums paid by them amount- 
ed to $300,453.85. We see from this, that the losses by death during 
the year exceeded the net annual premiums by more than $2,000,000. 
The "deposit" is reduced to $11,421*256, which is less than one half 
the amount in "deposit" at the end of the thirty-fourth year. But 
the company has not lost money, it has only been paying its debts. 
At the end of the thirty-fourth year it had more, but it owed more- 
It had enough then, and only enough, to pay what it owed ; it is in 
the same condition now. 

At the end of the sixty-fifth year, we find the " lef : sit " that must 



NOTES ON LIFE INSURANCE. 47 

be in the hands of the company to the credit of each policy is 
84560.87 ; and there are twenty of the original policy-holders living. 
The aggregate "deposit" for these twenty outstanding policies is 
$91,217.40. The $27,838,632.25 that the company had on hand at 
the end of the thirty-fourth year is now reduced to less than 
8100,000. But the company has only been paying its debts to 
policy-holders — not losing money. In fact, it had none to lose of 
its oicn. 

At the end of the sixty-ninth year, the "deposit" amounts to 
$4722.84 ; and there is one policy-holder living. He pays his regular 
net annual premium the day he is ninety-nine years old. The pre- 
mium is $84.85. This, added to the " deposit " on hand at the end 
of the preceding year, makes $4807.69 of this policy-holder's money 
in the hands of the company the day the policy-holder is ninety-nine 
years old. At net interest, which is four per cent, the interest for 
the year will amount to $192.31 ; and this, added to the amount, 
$4807.69, on hand at the beginning of the year, makes $5000, with 
which to pay the policy of the last policy-holder in this company. 

We see that the $27,838,632.25, which the company had in its 
possession at the end of the thirty-fourth year, belonging to policy- 
holders, has been paid to them. The policies were all paid at matu- 
rity ; the company has nothing left. In fact, it never had a cent of 
its own during the whole time, although we have seen it the custo- 
dian, at one time, of nearly twenty-eight millions of dollars of other 
people's money. It owed every cent, and it paid every cent it owed. 

It is a marked peculiarity of life insurance business, as seen in this 
illustration, that the annual premiums exceed the death-claims for 
the first thirty or forty years ; after which time, the losses by death 
largely exceed the annual premiums. The trust fund deposit, after 
a table of mortality and rate of interest have been designated, is a 
fixed mathematical amount ; it increases for each policy at the end 
of every succeeding year of the existence of the policy. 



IS 



:":r_5 :•>* luz ixs usance. 



Deposit at the end of each Year on a WhoU-IAfe JPoUcy for $1<XXH 

B:r.^ <:f I U -±s1. 



ATi. 


.r : _r _ -f-r : :_:.. 


percent. 


-.7 :i: It-:. 


?iz :■:: :-.r~:.. 


i*s 


^ 


J':"*?? 


$:-: :: 


$15.18 


$1: 42 


,. 


46 


>4 . .-* 


•:: ?" 


s: r\ 


•:-.4i 


L- 


47 


K 2 


-. :: 


4" il 


4: £ 


2f 


4- 


71 ^: 


r s: 


-T4 14 


:- C5 


£i 


-i 


ri :i 


s: :t 


*: 35 


"•: f4 


ii 


•:•: 


::: "■: 


1:4 ~f 


^ -:: 


Sr :v 


=,- 


f: 


is: ""-- 


1J: rl 


::- :.-; 


::••? sf 


=-' 


-:-: 


:-: : 


141 47 


l£i: ^ 


11; 15 


-■; 


*.- 


:-- fi 


: I 4: 


lie 4r 


14. ro 


='? 


51 


:S s' 


is: --: 


:~ > 


l!r 1? 


e^i 


55 


£14 if 


3:4.5- 


:^: :- 


it: h: 


=.-, 


56 


iS: ": 


ss a 


=1: --'. 


:-- .4 


:•:' 


5" 


£7 :.: 


24: :4 


i:.. :e 


CI-: :.- 




5* 


ST.- :•: 


a*- :•: 


•ifr r*r 


^:r 4^ 


-,; 


"- 


i::: ->: 


iS.- :c 


sr? :: 


Si B 


-i 


■..: 


S£ f> 


in 45 


i'r :•: 


i- C7 


-,• 


61 


E-4: 1-5 


i.:;; >s 


a s 


s* :7 


r-_ 


61 


h m 


>.: " S7 


C4i -r 


i:r :■: 


^ 


H 


y.?'. ~i 


i— >s 


>::4 S 


:^: cs 


:> - 


CI 


411 rl 


iVr :" 


396.30 


y ; : v 


h '4 


•rc 


414 * 


42' cs 


4.^ A 


c& -: 


if;' 


66 


4-V 32 


44; :e 


-.2r . ?: 


4:4 i ; 


0: 


r 


4-' 4: 


4:4 :> 


4:: i: 


44: -0 


: - 


* 


499.78 


4r- r * 


4-: -.; 


44: SJ 


r',i 


59 


"a: ~» 


.-.:■- :•:■ 


4-:; :.- 


4:- r'"J 




:: 


f4l .-S 


;.:- f* 


!14 >": 


4* :2 


7; 


71 


v. :.f 


-4- :: 


!-:4 "4 


:.:•: 1: 


-- 


89 


"fi >:- 


C~ r- 


:;:4 n; 


:l:- 4; 


~i 


" : , 


eo: s r 


:.:•'- 5" 


c"4 :S 


54.- ;: 


:i 


74 


55: :* 


:.:- :-2 


'< u: 


v:r 44 


74 


R 


-.;> ^ 


635.53 


f::E :S 


:* 57 


75 


7: 


■?." 7. 


•.4-f -^ 


■■'. : : : v 


■ri.-!.- > 


7i 


rs 


"f 5: 


"r4 15 


rx.i J" 


:-!- If 


n 


7; 


K-A ^ 


r;-S m 


-iT. r 


:4f :.: 


~5 


79 


r.c - 


r:: 4- 


«•: *:• 


:. ; .- :: 


79 


s: 


730.56 


-:.- -•= 


7> rS 


--'" 4, 


n 


i"! 


74f :r: 


~r v 


■827.17 


7> :.i 


f: 


ifi 


7:i: 14 


7:/ -15 


-4: :1 


735.64 


a 


SI 


7si r 


— : =& 


":.: 44 


"44 ~: 


>-- 


^ 


TS9.49 


7>. :.: 


7rl :4 


T:.i .-1 


-: 


S: 


fl: 41 


s :* :- 


~' J r "3 


?s :* 


fo 


S- 


s::; ?: 


g25.13 


?:: S7 


s:: s 


BE 


•c 


•4 » :-5 


^41 :4 


S:4 :- 


?:; r? 


«'- 


* 


s-,: -ji 


=&: :~ 


5&: :t 


SV SL 


s- 


rS 


i — . '4 


«-: :: 


Sv -^ 


v.: 4: 


f'.r 


N 


^4 -V 


SS ; 13 


ss: :T 


870.0* 


?: 


91 


^-■r s: 


Vr4 -: 


SS: Sf 


■rl 


93 


910.9S 


*>. 1- 


f:?T :if 


>: 


*: 


"rSi: 4'. 


>:-. :: 


-I" !.r 


-0 :■ .. 


:.i 


^ 


• : :; 4 4-: 


-::C ^: 


>:- 4= 




'^4 


95 ; 


1:0: :•: 


1000.00 


::.:»: :»: 


::•:•: :•: 





NOTES ON LIFE INSURANCE. 



49 



Deposit at the end of each Year on a "Whole-Life Policy for $1000, 
taken out at age 45 — Actuaries' — Various Hates of Interest. 



Age. 


Four per cent. 


Four and a half 
per cent. 


Five per* cent. 


Six per cent. 


Age. 


45 


$18.01 


$16.97 


$16.01 


$14. 2S 


45 


46 


36.36 


34.32 


32.42 


29.00 


46 


47 


55.04 


52.03 


49.22 


44.15 


47 


48 


74.03 


70.09 


66.40 


59.72 


48 


49 


93.34 


88.50 


83.96 


75.90 


49 


50 


112.94 


107.23 


101.87 


92.11 


50 


51 


132.80 


126.27 


120.12 


108. 90 


51 


52 


152.91 


145.60 


138.70 


126.08 


52 


53 


173.24 


165.19 


157.59 


143.62 


53 


54 


193.79 


185.05 


176.77 


161.53 


54 


55 


214.53 


205.14 


196.23 


179.78 


55 ' 


56 


235.48 


225.44 


215.94 


198.36 


56 


57 


256.50 


245.95 


235.91 


217.27 


57 


58 


277.70 


266.65 


256.11 


236.49 


58 


59 


299.01 


287.51 


276.51 


255.99 


59 


60 


320. 35 


308.44 


297.04 


275.70 


60 1 


61 


341.69 


329.44 


317.67 


295.60 


61 ' 


62 


362.99 


350.43 


338.35 


315.63 


62 


63 


384.21 


371.39 


359.04 


335.76 


63 


64 


405.30 


392.27 


379.56 


355.94 


64 


65 


426.23 


413.03 


400.28 


376.12 


65 


66 


446.93 


433.63 


420.74 


396.27 


66 


67 


467.39 


454.02 


441.04 


416. 33 


67 


68 


487.58 


474.18 


461.14 


436.28 


68 


69 


507.49 


494.10 


481.05 


456.10 


69 


70 


527.09 


513.74 


500.72 


475.75 


70 


71 


546.45 


533.08 


520.12 


495.21 


71 


72 


565.24 


552.09 


539.22 


514.44 


72 


73 


583.77 


570.76 


558.02 


533.43 


73 


74 


601.90 


5S9.07 


576.49 


552.14 


74 


75 


619.63 


607.01 


594.61 


570.56 


75 


76 


636.95 


624.56 


612.37 


588 68 


76 


77 


65385 


641.71 


631.67 


606.46 


77 


78 


670.29 


658.42 


646.72 


623.88 


78 


79 


686.30 


674.72 


663.29 


640.93 


79 


80 


701 89 


690.63 


679.48 


657.65 


80 


81 


717.13 


706.19 


695.36 


674.08 


81 


82 


732.20 


721.50 


710.99 


690.31 


82 


83 


746.85 


736.61 


726.45 


706.41 


83 


84 


761.48 


751.63 


741.84 


722.49 


84 


85 


776 00 


766.56 


757.15 


738.54 


85 


86 


790.42 


781.39 


772.40 


754.58 


86 


87 


804.73 


796 16 


787.61 


770.61 


87 


88 


818.87 


810.77 


802 67 


786.56 


88 


89 


832.72 


825.09 


817.47 


802.26 


89 


90 


846.24 


839.12 


831.98 


817.71 


90 


91 


859.31 


852.68 


846.03 


832.72 


91 


92 


871.68 


865.54 


859.37 


847.00 


92 


93 


883.10 


877.42 


871.71 


860.26 


93 


94 


893.36 


888.12 


882.84 


872.23 


94 


95 


901.61 


896.72 


891.79 


881.88 


95 


96 


907.99 


903.37 


898.72 


889.34 


96 


97 


916.51 


912.27 


907.99 


899.35 


97 


98 


932.69 


929.21 


925.68 


918.56 


98 


99 


1000.00 


1000.00 


1000.00 


1000.00 


99 



Note. 
cussion. 



-For further illustration of the manner of calculating the deposit, see Algebraic Dis- 



50 NOTES OK LIFE INSURANCE. 



CHAPTER IV. 

AMOUNT AT RISK— VALUATION OF POLICIES. 

Having obtained the amount that must be in deposit at the end 
of any year, subtract this from the amount called for by the policy, 
and we have the amount the company has at risk during that year. 
The deposit increases from year to year ; therefore on any given 
policy, the amount the company has at risk diminishes each year. 
During the last year of a policy that continues to the limit of the 
term of insurance, the amount the company has at risk is zero ; be- 
cause the deposit on hand at the beginning of the year, added to 
the net annual premium paid at that time, will, when increased by 
net interest for one year, produce the amount of the policy. 

In case of insurance for one year only, no provision is made in the 
net premium for a deposit at the end of the year. The amount at 
risk in this case is expressed by the face of the policy. At the 
younger ages, the net premium that will effect insurance for one 
year is quite small in comparison with the policy it will pay for. 
For instance, at age 20, the amount that will, if paid at the begin- 
ning of the year, insure $1, to be paid to the heirs of the insured, at 
the end of the year, in case he dies during the year (Actua- 
ries' 4 per cent), is equal to T ^ X drUs = $0.0070104. For a po- 
licy of $1000, the amount required is $7.01. At age 99, the amount 
requisite to insure $1 for one year is T .^ T X t = t-Vt — $0.961538. 
Therefore the amount that will at age 99 insure $1000 for one year 
is $961.54. In case the insured pays at the beginning of any year, 
a net amount greater than that necessary to effect the insurance dur- 
ing the year, the company, after setting aside the amount that will 
pay for the insurance, will have in its hands a certain portion of the 
policy-holder's money, and the amount the company actually has at 
risk during the year is not the full amount called for by the policy ; 
because the company will, in part payment of the policy, in case the 
insured dies during the year, use the money it holds belonging to 
the policy-holder. For instance, at age 20, a whole-life policy of 
$1000 is paid for by a net single premium, $251,907 (Actuaries' 
4 per cent, see table, page 26). This will effect the insurance during 



NOTES ON LIFE INSURANCE. 51 

the first year, and leave in the hands of the company at the end of the 
year, if the insured is living, an amount sufficient at that time, age 
21, to pay for insurance for whole life ; but we know from direct 
calculation (see same table) that the net single premium that will, 
at age 21, insure $1000 for whole life is $256,564. Therefore, when 
the insured paid $251,907 at age 20, he not only paid for insurance 
during the first year, but his own money supplied in addition $256.- 
564 at the end of the year. The amount the company has at risk 
during the year is $1000 less $256,564 = $743,436. The amount 
that will, at age 20, insure $1000 for one year is $7.0104 (see table, 
page 19). The amount required at same age to insure $743,436 for 
one year is found from the proportion $1000 : $743,436 : : $7.0104 : 
85.211. Therefore $5,211 is the amount that will, if paid at the be- 
ginning of the year, effect insurance during the year on the $743.- 
436 that the company has at risk that year. As a test of this, no- 
tice that the net single premium paid at age 20, namely, $251,907, 
will, when increased by interest during the year at 4 per cent, 
amount to $261,983 ; but $5,211 of the net premium paid at the 
beginning of the year was for the purpose of insuring the amount 
at risk during the year. This, increased by 4 per cent, gives at the 
end of the year $5,419, to pay net cost of insurance at the end of 
the year on $743,436, at risk during the year. If Ave subtract $5,419 
from $261,983, the remainder ought to give the net single premium 
at age 21. We have $261.983 —$5,419 = $256,564. The latter 
amount is, as previously shown by direct calculation for each sepa- 
rate year, the net single premium forage 21. In like manner, when 
the deposit for the end of the second year has been calculated, by 
subtracting this from the amount of the policy, we obtain the 
amount at risk during the second year. The net single premium 
that will at age 22 insure $1000 for whole life, as previously obtain- 
ed by direct calculation for that age, is $261,378 (see table). This 
is the amount that must be on deposit to the credit of this policy, 
in case the insured is alive at age 22. 

The amount at risk during the second year is, therefore, $1000 — 
$261,378 = $738,622. We have previously determined by direct cal- 
culation (see table, page 19), that at age 21 the amount that will, if 
paid at that age, insure $1000 for one year is $7,093. Therefore we 
obtain the amount that will, if paid at age 21, insure the amount the 
company has at risk the second year by the proportion $1000 : $738.- 
622 : : $7,093 : $5,239. Therefore, $5.239 is the amount that will, if 
paid at the beginning of the second year, insure the amount the com- 
pany has at risk during that year. Increase this by 4 per cent for 



52 XOTES ON LIFE IXSUEAXCE. 

one year, and we have $5,449, which will pay at the end of the se- 
cond year the cost of insurance on the amount at risk during that 
year. The amount in deposit at the beginning of the second year is 
$256,565. This, increased by interest at 4 per cent for one year, 
gives $268,827, with which, at the end of the year, to pay cost of 
insurance on amount at risk during the second year, and leave in 
the hands of the company, in deposit, an amount that will, at age 
22, effect the insurance of the policy for life. Therefore, after cost 
of insurance the second year is paid, we have $266,827 — $5,449 = 
261.379 in deposit at the end of the second year. This is the 
amount that will, at age 22, insure $1000 for life, as determined by 
direct calculation, for each separate year to the table limit. 

In a manner similar to the above, the account of this policy can 
be carried year by year to the end of the table, and it will be found 
that after cost of insurance on the amount at risk each year has 
been paid, there will be on deposit at age 99 an amount which, at 
4 per cent, will produce $1000 in one year. 

"VTe will now assume that a policy for $1000 is issued at age 42, 
and paid for by net annual premiums. It has already been shown 
that the net annual premium (Actuaries' 4 per cent) which will in- 
sure this policy is $25,554 (see table, page 33). When the first pre- 
mium is paid, it not only effects the insurance during the first year, 
but it provides a deposit for this policy at the end of the year. 
The amount that must be in deposit for this policy at age 43 is 
(see page 44) $15,851. Therefore the amount the company has 
at risk during the year is $1000— $15.851— $984,149. By direct 
calculation, it is found that the amount that will, if paid at age 42, 
insure $1000 for one year is $10,476 (see page 19). We obtain the 
amount that will, if paid at the same age, insure $984,149 by the 
proportion $1000 : $984,149 : : $10,476 : $10.31. Increase $10.31 at 
4 per cent for one year, and we have $10.72 at the end of the year 
to pay cost of insurance on $984,149, the amount the company had 
at risk during the year. The net annual premium $25,554, paid at 
the beginning of the year, will, when increased by 4 per cent, 
amount to $26.57 at the end of the year. Subtract from this the 
cost of insurance on the amount at risk during the year which, as 
found above, is $10.72, and the remainder $15.85 is equal to the ex- 
act amount that by an independent calculation is known to be 
the deposit that must be in the hands of the company, to the credit 
of this policy at age 43. 

It has been previously seen (page 44) that for the second year of 
this policy, the amount that must be on deposit at the end of this 



NOTES ON LIFE INSURANCE. 53 

year is $32.17236. The amount the company has at risk during the 
second year is therefore equal to $1000— $32.17236=$967.82764, 
and this is the amount of insurance the policy-holder gets from the 
company during the second year. The amount that will, if paid at 
age 43, insure $1000 for one year is $10,818 (see table, page 19). 
To obtain the amount that will, if paid at the same age, insure 
the amount at risk during the year, we use the proportion $1000 : 
$967.82764 :: $10,818 : $10.47. 

Therefore $10.47 is the amount that will, if paid at age 43, insure 
the amount the company has at risk during the second year of this 
policy. Increase this by 4 per cent for one year, and we have 
$10,889, which is the amount in the hands of the company at the 
end of the second year to pay the cost of all the insurance the pol- 
icy-holder has had from the company during that year. The depo- 
sit on hand at the end of the first policy year was $15.85. The net 
annual premium paid at the beginning of the second year was $25.55. 
Therefore the company had in its possession at the beginning of the 
second year, $41.40 of this policy-holder's money to the credit of 
this policy. This increased by 4 per cent amounts at the end of the 
year to $43.06, out of which must be paid $10,889, the cost of in- 
surance during the year, which leaves on hand $32.17, the deposit 
for the end of this year, as shown by previous direct calculation. 

We have just seen that $10.47 will, if paid at age 43, insure the 
amount at risk on this policy during the year between age 43 and 
age 44. This amount at that time pays for insurance for one year on 
$967,828. The first question is, what is the value of this payment 
at age 42, when this policy was issued? $10.47 to be paid certain 
in one year, interest being at the rate of 4 per cent per annum, is 
equal to y.f^ X $10.47 ; but it is only to be paid if the insured is 
alive at age 43. From the table of mortality (Actuaries') we find 
that the fraction which at age 42 represents the chance that the in- 
sured will be alive at age 43 is fffcff- Therefore, T ^ X $10.47 X 
j ? o 1 1 expresses the value at age 42 of the cost of insurance on the 
amount at risk during the year between age 43 and age 44. 

In like manner, at the beginning of each year of the term of the 
policy, find the amount at risk during that year ; find the amount 
that will, at the beginning of the year, pay, at the end of the year, 
the cost of insuring the amount at risk that year ; find the amount 
that will, if paid at age 42, produce at the beginning of the year in 
question the amount that must then be paid to provide for cOst of 
insurance during that year, and multiply the result by the fraction 
which at age 42 represents the chance that the insured will be alive 



54 



NOTES ON LIFE IXSUEANCE. 



at the beginning of the year for which the calculation is made ; add 
together all these respective yearly amounts, and we have the 
amount that will, if paid at age 42, effect insurance on the amount 
the company has at risk on this policy of $1000. 

From the foregoing remarks, it is seen that the net premium is 
composed of two parts, one of which, with net interest thereon, goes 
to pay each year the cost of insurance on the amount the company 
has at risk during that year ; the other, with net interest thereon, 
goes to form the deposit or reserve that must be held at the end of 
each year to the credit of the policy. After a number of years, it 
sometimes happens that the whole of the net annual premium, with 
net interest thereon, is not sufficient to pay cost of insurance on the 
amount at risk during the year ; but in all such cases, the deposit at 
the beginning of the year, with net interest thereon, will be sufficient 
to provide the requisite deposit at the end of the year, and make 
up whatever deficiency there may be in the net annual premium at 
net interest, in paying cost of insurance on the amount at risk dur- 
ing the year. 

ILLUSTRATIVE TABLE. 

Whole-Life Policy for 81000 issued at Age 20, paid for by equal 
Annual Premiums of $11,966 each — American Experience, four 
and a half per cent. 





Xet value in- 
Net value at the creased by four 


Amount at 
risk. 


Cost of in- 
surance on 
amount at 
risk. 


Deposit or re- 




Age. 


beginning of the 
year. 


and a half per 
cent during the 


serve at the end 
of the year. 


Age. 






year. 










20 


811.966 


$12,504 


$995,264 


$7,768 


$4,736 


20 


21 


16.702 


17.454 


990.325 


7.779 


9.675 


21 


22 


21.641 


22.615 


985.175 


7.790 


14.825 


22 


23 


26.791 


27.997 


979.801 


7.798 


20.199 


23 


24 


32.164 


33.612 


974.193 


7.804 


25.807 


24 


25 


37.773 


39.473 


968.336 


7.809 


31.664 


25 


26 


43.630 


45.593 


962.230 


7.823 


37.770 


26 


27 


49.736 


51.974 


955.861 


7.835 


44.139 


27 


28 


56.105 


58.629 


949.216 


7.845 


50.784 


28 


29 


62.750 


65.574 


942.290 


7.864 


57.710 


29 


30 


69.676 


72.811 


935.068 


7.879 


64.932 


30 


31 


76.S9S 


80.358 


927.356 


7.894 


72.464 


31 


32 


84.430 


88.229 


919.687 


7.916 


80.313 


32 


33 


92.279 


96.431 


911.515 


7.946 


88.485 


33 


34 


100.451 


104.971 


903.003 


7.974 


96.997 


34 


35 


108.963 


113.866 


894.133 


7.999 


105.867 


35 


36 


117.&33 


123.135 


884.907 


8.042 


115.093 


36 


37 


127.059 


132.776 


875.307 


8.0S3 


124.693 


37 


38 


136.659 


142.809 


865.333 


8.142 


134.667 


38 


39 


146.633 


153.231 


854.965 


8.196 


145.035 


39 


40 


157.001 


164.066 


844.203 


8.269 


155.797 


40 


41 


167.763 


175.313 


833.024 


8.337 


166.976 


41 


42 


178.942 


186.994 


821.428 


8.422 


178.572 


42 


43 


190. 538 


199.112 


809.400 


8.512 


190.600 


43 


44 


202.566 


211.682 


796.949 


8.631 


203.051 


44 


45 


215.017 


224.692 


784.060 


8.752 


215.940 


45 


46 


227.906 


238.161 


7/0.750 


8.911 


229.250 


46 



NOTES ON LIFE INSUKANCE. 



55 





1 


1 Net value in- 




1 Cost of in- 








Net value at the creased by foui 


Amount at 
risk. 


surance on 


Deposit or re- 




Age. 


beginning of the 


and a half per 


amount at 


serve at the end 


Age. 




year. 


,ceiit during the 


risk. 


of the year. 








year. 










47 


$241,216 


$252,071 


$757,014 


$9,085 


$242,986 


47 


48 


254.952 


266.425 


742.868 


9.293 


257.132 


48 


49 


269.098 


281.207 


728.339 


9.546 


271.661 


49 


50 


283.627 


296.390 


713.442 


9.832 


286.558 


50 


51 


298.524 


311.957 


698.195 


10.152 


301.805 


51 


52 


313.771 


327.891 


682.613 


10.504 


317.387 


52 


• 53 


329.353 


344.173 


666.717 


10.890 


333.283 


53 


54 


345.249 


360.785 


650.531 


11.316 


349.469 


54 


55 


361.434 


377.698 


634.077 


11.775 


365.923 


55 


56 


377.889 


394.894 


617.383 


12.277 


382.617 


56 


57 


394.583 


412.339 


600.472 


12.811 


399.528 


57 


58 


411.494 


430.011 


583.369 


13.380 


416.631 


58 


59 


428.597 


447.884 


566.110 


13.994 


433.890 


59 


60 


445.856 


465.918 


548.729 


14.647 


451.271 


60 


61 


463.237 


484.083 


531.260 


15.343 


468.740 


61 


62 


480.706 


502.337 


513.739 


16.076 


486.261 


62 


63 


498.227 


520.647 


496.195 


16.842 


503.805 


63 


64 


515.771 


538.380 


478.670 


17.650 


521.330 


64 


65 


533.296 


557.294 


461.214 


18.508 


538.786 


65 


66 


550.752 


575.536 


443.864 


19.400 


556.136 


66 


67 


56S.102 


593.666 


426.663 


20.329 


573.337 


67 


68 


585.303 


611.641 


409.662 


21.303 


590.338 


68 


69 


602.304 


629.407 


392.894 


22.301 


607.106 


69 


70 


619.072 


646.930 


376.404 


23.334 


623.596 


70 


71 


635.561 


664.161 


360.213 


24.374 


639.787 


71 


72 


651.753 


681.082 


344.304 


25.386 


655.696 


72 


73 


667.662 


697.706 


328.644 


26.350 


671.356 


73 


74 


683.322 


714.071 


313.184 


27.255 


686.816 


74 


75 


693.782 


730.237 


297.885 


28.112 


702.115 


75 


76 


714.081 


746.215 


282.710 


28.925 


717.290 


76 


77 


729.256 


762.072 


267.655 


29.727 


732.345 


77 


78 


744.311 


777.805 


252.732 


30.537 


747.268 


78 


79 


759.234 


793.400 


237.946 


31.346 


762.054 


79 


80 


774.020 


808.851 


223.427 


32.278 


776.573 


80 


81 


788.539 


824.023 


209.149 


33.172 


790.851 


81 


82 


S02.817 


838.944 


195.053 


33.997 


804.947 


82 


83 


816.913 


853.674 


180.999 


34.673 


819.001 


83 


84 


830.967 


868.361 


166.919 


35.280 


833.081 


84 


85 


845.047 


883.074 


152.955 


36.029 


847.045 


85 


86 


859.011 


897.666 


139.359 


37.025 


860.641 


86 


87 


872.607 


911.874 


126.440 


38.314 


873.560 


87 


88 


885.526 


925.374 


114.227 


39.601 


885.773 


88 


89 


897.739 


938.137 


102.299 


40.536 


897.601 


89 


90 


909.567 


950.497 


90.755 


41.252 


909.245 


90 


91 


921.210 


962.665 


79.856 


42.521 


920.144 


91 


92 


932.110 


974.055 


70.940 


44.995 


929.060 


92 


93 


941.026 


983.372 


62.550 


45.922 


937.450 


93 


94 


949.416 


992.139 


55.028 


47.167 


944.972 


94 


95 


956.938 


1000.000 


00.000 


00.000 


1000.000 


95 



Valuation of Policies. — At the time the first premium is paid, 
which is at the beginning of the first policy year, the net value of the 
policy is the net annual premium. At the end of the first policy 
year, the net cost of insurance will have been paid, and there must 
be left in the hands of the company, in trust for the policy-holder, 
the requisite " deposit." This deposit (or " reserve" as it is often 
called) is the net value of the policy at the end of the first policy 
year. At the beginning of the second policy year, the net annual 
premium is paid, and the net value of the policy is then the " de- 
posit " at the end of the preceding year, plus the net annual pre- 
mium just paid. The net value of the policy at the end of the se- 



56 NOTES ON LIFE INSURANCE. 

cond policy year is the deposit (or " reserve") for the end of that year, 
and the net value of the policy at the beginning of the third policy 
year is equal to the deposit at the end of the second year, plus the 
net annual premium paid at the beginning of the third year. The 
rule is general, and applies to every year the policy is in force, be- 
cause the net annual premium is sufficient, and only sufficient, when 
added to the " deposit" at the end of the preceding year, to pay the 
net cost of insurance during the year, and provide the requisite de- 
posit for the end of the year. Of course it is understood that net or 
table interest is realized for the year. 

On the supposition that a policy was taken out on the 1st day of 
January, 1875, the net value of the policy on that day is equal to 
the net annual premium just paid. On the 31st December, 1875, 
the net value is equal to the " deposit" at the end of the first policy 
year. On the 1st day of January, 1876, the net value of the policy, 
just after the net annual premium is paid, is equal to the deposit at 
the end of the preceding year, plus the net annual premium ; and 
the net value on the 31st of December, 1876, will be equal to the 
deposit at the end of the second policy year. 

Having in this way determined the value of this policy at the be- 
ginning and at the end of any policy year, subtract one from the 
other, and by this means obtain the difference between the net value 
on the 1st day of January, and the net value on the 31st day of De- 
cember of that year. Divide this difference by twelve : we will ob- 
tain the monthly difference in the net value. Assuming that the net 
value of a policy is greater at the beginning than it is at the end of 
the policy year in question ; having found the monthly difference as 
above, we will subtract this monthly difference from the net value at 
the beginning of the year, in order to find the net value of this 
policy on the 1st day of February of that policy year. To find the 
net value of the policy on the 1st day of March, we will subtract the 
monthly difference from the net value on the 1st day of February ; 
and in like manner we obtain the net value of the policy at the be- 
ginning of any month of the policy year, by subtracting from the 
net value at the beginning of the policy year this monthly difference 
multiplied by the number of months of the policy year that have 
expired. 

On the 1st day of November, for instance, we obtain the net value 
by multiplying the monthly difference by ten, and subtracting the re- 
sult from the net value of the policy on the ] st day of January, 
which day we have assumed to be, in this case, the first day of the 
policy year. 



NOTES ON LIFE INSURANCE. 57 

To obtain the net value on any day during a month, divide the 
monthly difference by thirty, in order to obtain the daily difference ; 
and then use the daily difference in a manner entirely similar to that 
indicated above for finding the value of the policy at the beginning 
of any month. 

Policies are taken out any day of the year, and it is usual in life 
insurance companies to have the net valuation of all policies com- 
puted on some one day every year. The day fixed for these valua- 
tions is generally the 31st of December. 

The question will then arise every year, What is the net value, on 
the 31st of December, of each policy in force on that day ? 

First, determine what policy year the given policy is in at the time. 
Obtain its net value at the beginning of that policy year, and its net 
value at the end of that policy year. Take the difference between 
these two net values : divide this difference by twelve, in order to 
obtain the monthly difference in the net value ; divide the monthly 
difference by thirty, in order to obtain the daily difference in net 
value. Then fix the month and day of the calendar year on which 
the policy was issued. The number of months and days that have, 
on the 31st of December, elapsed since the beginning of the policy 
year, will become known, and the net value of the policy on the 31st 
day of December can be determined by the general method above 
indicated. 

We might make this calculation without reference to monthly 
differences by dividing the yearly difference by 365, in order to ob- 
tain the daily difference, and then multiplying this by the number of 
days from the beginning of the policy year in question, to the 31st 
day of December of that year. A table has been constructed show- 
ing the decimals of a year from each day to the 31st December in- 
clusive, which will facilitate the calculation. Using this table, we 
have only to multiply the yearly difference by the decimal of a year 
opposite the month and day on which the policy was issued. This 
gives the difference in net value Jbetween the beginning of the policy 
year and the 31st December following. 

If the net premium the company has agreed to receive is less than 
that called for by the designated data, then, since the deposit at the 
end of any policy year must be such an amount as will, when added 
to the value at that time of the net premiums still receivable, be 
equal to the net single premium at that time, it follows that when- 
ever a company makes contracts of life insurance at a rate of net 
premium less than that called for by the legal standard of safety, it 
must have at the end of a policy year, in deposit, in addition to 



58 NOTES ON LIFE INSURANCE. 

the net value above determined, an amount equal to the value at 
that time of a series of net premiums each of which is equal to the 
difference between the net premium called for by the standard and 
the net price the company has agreed to receive. 

Owing to some jDeculiarity in the rate of mortality for the year, 
and the accumulation of interest arising from the funds on deposit, 
it happens at times, especially in whole-life policies paid for by equal 
annual premiums, that the net value of a policy, at the beginning of a 
year, will, at net interest, produce, at the end of the year, an amount 
sufficient to pay the cost of insurance during the year, and provide 
for a u deposit " at the end of the policy year, greater than the net 
value at the beginning of the yeai\ In this case, the monthly and 
daily differences must be added to the net value at the beginning of 
the year, instead of being subtracted from it. This peculiar case 
does not happen in the earlier years of a policy ; it is only after 
there is marked accumulation in the " deposit " or net value at the 
end of a year, that the net value at the beginning of a year will, 
at net interest, produce an amount sufficient to pay the cost of 
insurance during the year, and leave on hand at the end of the year 
a " deposit," or net value, greater than that at the beginning of the 
year. 

These "perturbations " in the relative net values at the beginning 
and end of different years are indicated in the formula by unmistak- 
able signs ; they in no degree complicate the calculations, but re- 
quire close observation on the part of computers to prevent mistakes. 

The net value, at any time during a policy year, can be obtained 
with equal certainty by basing the calculations upon the deposit or 
net value of the policy at the end of the policy year, instead of, 
as above, upon the net value at the beginning of the j^olicy year. 

Having calculated the deposit that must be on hand at the end of 
the policy year, the value of the policy at the end of the first month 
of the policy year may be obtained by adding to the deposit that 
must be on hand at the end of the ^ear eleven twelfths of the cost 
of insurance during the year. At the end of the second month the 
net Value may be obtained by adding to the deposit that must 
be on hand at the end of the year ten twelfths of the cost of insur- 
ance during the year. At the end of the eleventh month, one twelfth 
is added. At the end of the twelfth month, or end of the policy 
year, there is nothing to be added. The net value and the deposit 
at the end of the year are equal. 

What is said above in reference to the particular case in which the 
deposit or net value at the end of a policy year is greater than the 



NOTES ON LIFE INSURANCE. 59 

net value at the beginning of the year, applies here ; and, therefore, 
j when the case occurs, the eleven twelfths of the difference between 
j the net value at the beginning and that at the end of the year 
must be subtracted from the net value or deposit at the end of the 
year, in order to obtain the net value at the end of the first month 
of the policy year ; and in like manner for other months. 

It is assumed in both of the methods for calculating the net value 
of a policy during the policy year, that the variation in value is pro- 
portional to the time, and that each month has thirty days. 

The net values of many of the different kinds of policies on the 
31st of December, in each policy year, have been calculated and ar- 
ranged in Valuation Tables convenient for use. Without the aid 
of these " Valuation Tables," the work of computing the net value of 
every policy in all the companies would be an almost impracticable 
labor. Even with the aid of " Valuation Tables," the work is enor- 
mous, as may be readily comprehended from the fact that one 
single company has more than ninety thousand policies in force. 



60 NOTES OX LIFE INSURANCE. 



CHAPTER Y. 

JOINT LIVES. 

Note 1. If there be a chances of the happening of any event, that mnst either happen or 
fail to happen, and b chances for its not happening, then Mill the probability of ench event 

taking place be represented by - — — • The probability of the event not happening -will be 

expressed by- — — -• The sum of these two fractions representing the probabilities of the 

happening and the failing is equal to unirv. because — — - — - — — - = — — — = 1. From 

a-t-b a-j-b a -J- b 

this it follows that, one of the two fractions being given, by subtracting this from unity, we 

will obtain the other fraction.— {Doctrine of Chances.) 

Xote 2. In case we have to determine the fraction which represents the chance that two 
events will happen, it is necessary first to find the fraction that represents the chance in each 
separate case, and then multiply one of these fractions by the other. Suppose that there are 
two boxes, each containing 100 balls, 99 of which are black and one is white ; the chance of the 
white ball being drawn from the first box is one out of 100 and is expressed by the fraction 
x^;. The person who drew the white ball from the first box now draws from the second box ; 
again, his chance of drawing the white ball is only one out of one hundred, expressed by ^. 
The chance of his getting both white balls is equal to the one hundredth part of ?£& or equal to 
■j^ x Tir<j = l oooo ' The same method of reasoning may be applied to the happening of three 
or any other number of events. — (Doctrine of Chances.) 

fVet Single Premium — Joint Lives. — By the terms of an in- 
surance contract upon two joint lives, the condition usually is that 
the policy is to be paid to the survivor at the end of any year dur- 
ing which either of the two joint lives may fail. There is marked 
similarity between the method of calculating the net single pre- 
mium for insurance on joint lives and that already explained for 
calculating the net single premium in case one life onlv is insured. 

OCX •. 

Tiro Joint Lives each aged, 40. — The fraction which represents the 
chance that a person aged 40 will live until he is 41 is expressed 
by 77341 divided by T8106 ; as shown by the American Experience 
Table of Mortality. When we add the condition that another per- 
son aged 40 will live until he is 41, the fraction which expresses 
the chance that the two lives will continue in being together dur- 
ing the first year is expressed by j|f$| X ^ I f o I- Consequently 
the fraction which represents the chance that these two joint lives 
will not continue in being together through the first year is express- 
ed by 1 — \\\%\ X -I jo} - ^ ne am(nmt tnat wnl - at 4 T P er cent 
produce $1 certain in one year is expressed by x.Jrg. Multiply this 
by 1 — ?8iol X ?»ioli an d tne result will give the amount that 
will insure these two joint lives for the first year. 



NOTES ON LIFE INSURANCE. 61 

The amount that will at age 40 insure these two joint lives dur- 
ing the second year is equal to (t.-oV?) 2 , multiplied by the fraction 
which at age 40 represents the chance that the joint continuance 
of the two lives will cease during that year. At the beginning of 
the second year — that is, at age 41 — the fraction which expresses at 
that time the chance that these two lives will continue in being to- 
gether until age 42 is expressed by -ff-fi -} X -HfH- Therefore 
the chance at age 41 that these two lives will not continue in being 
together until age 42 is expressed by 1 — T f Jj-J- X ??%!? ■ Multi. 
ply this by the chance at age 40, that the two lives will continue in 
being together until age 41, and we have : 
( _ <76567 x 1Q5Q7 \ / ?7341 x 77341\ = 1 _ 76567 X 76567 



77341 77341/ ^78106 78106/ 78106 X 78106 

The last expression gives the chance that the continuance of the 
two joint lives will be interrupted during the second year. There- 
fore the amount that will at age 40 insure these two joint lives dur- 
ing the second year is expressed by : 

/ 1 V f 76567 X 765 67 i 
U045/ [ 78106 X 78106/ 

In like manner, obtain the fraction which at age 40 repre- 
sents the chance that the continuance of the two joint lives will 
be interrupted during the third year, by first finding the fraction 
which at age 42 represents the probability that the two lives 
will continue in being together until age 43. Subtract this fraction 
from unity. This gives the probability, at age 42, that the two 
lives will not continue in being together during the third year. 
Multiply this by the fraction which at age 40 represents the chance 
that the two lives will continue in being together until age 42, and 
we have an expression for the fraction which at age 40 represents 
the probability that these two insured lives will not continue in 
being together during the third year. Multiply (y.^^) 3 by this 
fraction, and we have the amount that will at age 40 insure these 
two joint lives for the third year. 

This being done for each year to the table limit, the sum of all 
these yearly amounts gives the net single premium that will at age 
40 insure these two joint lives. 

The same principles apply to joint lives of unequal ages, and to 
any number of joint lives. 

The Value of a series of Annual Payments of $1 each, on condi- 
tion that two Joint Lives continue in being together. — This is calcu- 
lated as follows : The first payment is made in hand, and it is $1. 



62 NOTES ON LIFE INSURANCE. 

The second is to be made at the beginning of the second year, pro- 
vided the two joint lives are in being at that time. In case of two 
joint lives aged 30 and 35 respectively, the fraction which repre- 
sents the chance, at the time the insurance is effected, that the two 
joint lives will be in being together at the beginning of the second 
year, is expressed by f§ £f| X fftM- Multiply T ^ by the last ex- 
pression, and we obtain the value in hand of the second payment of 
this series. In a similar manner, the value of each payment of the 
series may be calculated. The sum of the values of all these re- 
spective yearly payments will be the value of the whole series. 

The net annual premium that will insure $1 on joint lives is ob- 
tained by dividing the net single premium that will effect the in- 
surance by the value of a series of annual payments of $1 each as 
above. 

Construction of Commutation Columns — Joint Lives- — Ameri- 
can Experience, 4\ per cent. — Begin at age 95. First, take the case 
of two joint lives, and assume that they are of equal age. By the 
table, all living at age 95 die before age 96 ; consequently there is 
no chance, according to this table, that the two joint lives will con- 
tinue in being together during the year ; therefore, the chance that 
they will not continue in being together during the year is equal to 
unity minus zero. There being, by the table, none living at age 
96, the fraction which at age 95 represents the chance that the in- 
sured will be alive at age 96 may be expressed by -f . The net sin- 
gle premium that will at age 95 insure $1, to be paid at the end of 
the year, provided either of the two joint lives fails during the year, 
is therefore expressed by y.-^g- X (1 — f X l)^ tvoVs"- Assum- 
ing that the number of these joint-life insurances is equal to the 
number living in the table at age 95, which is 3, they will all be in- 
sured by an amount expressed by 3 X t^t» an ^ eacn set °f joint 
lives will be insured by an amount equal to 

3y 1 _3 3 1 _ 3 X 3 X yjyg 

3 1045 3 3 1.045 3X3 

Multiply the numerator and denominator of this fraction by 

( T ^y\ This gives 8 X 8 X (t-tJ'st)" . Call the numerator of this 

3 X 3 X (t-oVt) 95 
fraction C 95 . 95 , and the denominator D 95 . 95 , and we have the net 
single premium at age 95 for these two joint lives expressed 

llV 9 59 5 . j^959_5 

y D ~D ' 

•^9595 9595 



NOTES ON LIFE IXSUKANCE. 63 

At age 94, the amount that will insure for one year two joint lives 
of this age is expressed by t .oVf multiplied by the fraction which 
expresses at age 94 the chance that the two lives will not con- 
tinue in being together during the year. Therefore we have T .-^-^ 
X (1 — WY X A) — tne amount that will at age 94 insure $1, to 
he paid at the end of the year in case either of these joint lives fail 
during the year. The expression may be written : 

21 Xt-ttV^X (1— JV X A) . 
21 

Multiply the numerator and denominator by (t/oVt) 94 ? anc * ** be- 
comes 21* (t.»Vt)" X (1 - & X A) = 9mi± 



94-94 



21 X ( t .*Vt)" D, 

The amount that will at age 94 insure these two joint lives during 

the second year is obtained by multiplying (y.-oV^) 2 ^v the fraction 
which at age 94 expresses the chance that the two joint lives will 
not continue in being together between age 95 and age 96. The 
fraction which at age 95 expresses the chance that the two joint 
lives will not continue in being together until age 96 has just been 
found to be unity. Multiply this by -^ X /t> which is the fraction 
at age 94 that represents the chance that the two lives will con- 
tinue in being together until age 95, and we have ^V X -rp This is 
the expression which at age 94 represents the chance that the 
joint continuance of the two lives will cease during the year 
between age 95 and age 96. Therefore (y.oVr) 2 X A X -£i = 

3 X 3 X ( * -V 

— _ — _ vi-04o/ — the amount that will at age 94 insure these 

21 X - ; 1 
joint lives during the second year. Multiply the numerator and 
denominator of the fraction by (yoVs") 94 > and we have 
3 X 3 X (t.oW 6 C, 



)5'95 



2! X 21 X (t.oVtt)" »,,„ 
Therefore — 94 ' 94 — ?^?a ii^ 9 * — net single premium that will at 

94-94 94-94 

at age 94 insure $1 for whole life on two joint lives, each aged 94. 

In a similar manner find the net single premium that will at age 
93 insure two joint lives of equal ages. The numerator in this case 
having been calculated, it will be found that the second term is 
identical with C 94 . 94 , and the third term with C 95>95 . Therefore we 
have only to compute C 93#93 and D 93 . 93 in order to obtain the net 
single premium for these two joint lives at age 93. Continue in 
this way at each successive younger age to the table limit, and we 
have the C, M, and D commutation columns for two joint lives of 



64: NOTES ON LIFE INSUKANCE. 

equal ages, from which the R, N, and S columns for joint lives of 
these ages can be easily made. 

When the difference of ages of two joint lives is one year, a set 
of commutation columns is constructed in a manner similar to the 
above. Assuming that the older of the two lives has reached the 
age of 95, the factor -f.ihji introduced into the numerator and deno- 
minator, is raised to the 95th power ; at age 94 to the 94th power, 
and so on each successive year, until the younger life is aged 10. 

When the difference of ages is two years, a set of columns is 
constructed for this case, and so on increasing the difference of 
ages one year, until the columns are completed for every combina- 
tion of two ages up to and including a difference in age of 85 years. 

When there is a difference in the ages of the two joint lives, the 
D column may be constructed by multiplying the D, for single 
lives, of the older, by the number living at the age of the younger 
— for instance, the two joint lives having a difference in age of 10 
years, commencing the calculation at age 95, D 95 . 85 will be formed 
by taking D 95 for single lives, and multiplying it by the number 
living at age 85. 

When three or more lives are associated in joint insurance, the 
same general principles apply. The commutation columns for joint 
lives are too voluminous for general publication. 



PART I.— CONTINUED. 



Algebraic Discussion. 



NOTES ON LIFE INSURANCE. 67 



CHAPTER VI. 

NET PREMIUMS. 

The arithmetical discussion of the method of calculating net val- 
ues in life insurance was commenced by giving the rule for determin- 
ing the amount of money that will, if invested at a given rate of in- 
terest per annum, compounded annually, produce $1 in any desig- 
nated number of years. 

As this is a subject of vital importance in these calculations, the 
algebraic discussion will be preceded by further remarks on com- 
pound interest. 

We will suppose that the amount to be placed at interest is unity, 
it may be 1 cent, 1 dollar, or any unit of value. We will assume 
that it is one dollar. The rate of interest is represented by r for any 
unit of time, 1 day, 1 month, 1 year, or any one defined length of 
time. Then in the given unit of time the $1 will produce an amount 
of interest equal to r ; and at the end of the first unit of time there 
will be on hand, adding interest to principal, an amount equal to 
$1 -f- r. This amount is to be placed at interest during the second 
unit of time. 

We have just seen that $1 will, in a unit of time, at a rate of in- 
terest r, produce the amount 1 -(- r. Any other sum placed at in- 
terest at the same rate and for the same length of time must produce 
a proportional amount. Hence, we have, $1 : 1 -J- r :: 1 -f- r is to 
the amount that will be produced by 1 -f- r in one unit of time. 
Therefore, 1 -|- r multiplied by 1 -j- r > or !+"?» 1S tne amount 
that will be produced in a unit of time by an amount 1 -f- r at the 
rate r. Then, since $1 will, in the same length of time, and at the 
same rate of interest, produce 1 -f- r, it follows that j~T~? is the 
amount that will be produced by $1 in two units of time when in- 
terest is compounded at the rate r. 

At the beginning of the third unit of time, the amount to be 
placed at a rate of interest r during that time is 7^77 2 ; from the pro- 
portion, 1 j 1 -)- r :: r+7 1 * s to tne amount at the end of the third 
unit of time, it follows that r+T* i s tne amount that will at the rate r 
be produced by $1 in three units of time, at compound interest. In 
like manner it may be shown that at the end of four units of time 



68 NOTES ON LIFE INSUKANCE. 

the amount produced is expressed by r+T*« In short, the rule is 
general : " First add the rate of interest to unity, and then raise 
this quantity to a power, the exponent of which is the number of 
units of time." 

In illustration, suppose the unit of value is $1, the rate of interest 
is 4 per cent per annum, the two added together make $1.04. For 
the end of the second year it is 7^ 2 , for the third ^ 3 , for the end of 
1000 years it is I^ 1000 , and so on for any named period. 

To find what $1 will amount to if placed at compound interest at 
4 per cent per annum, for 1000 years, we have by simple arithmetic 
to multiply 1.04 by itself 999 times. A much shorter and easier 
process would be to find the logarithm of the number 1.04, multiply 
this by 1000, and find the number corresponding to the logarithm 
obtained by this multiplication. The latter computation can be 
easily made in round numbers in a few minutes' time, and the result 
shows that $1 placed at compound interest, at the rate of 4 per cent 
per annum, will, in 1000 years, amount to $107,978,999,539,174,369. 
Now, let us suppose the question is, How much money will, if in- 
vested at the rate r, produce 1 unit of value in 1 unit of time ? We 
will again assume that the unit of value is $1. We have seen that 
$1 invested at the rate r will in 1 unit of time produce an amount 
1 -\- r ; now if 1 -f- r is produced by $1, the following proportion 
shows that $1 will be produced in the same time at the same rate of 
interest by an amount equal to T \-^. That is, 1 -\- r : 1 :: 1 : T +i- 
Now the question is, how much money will it require to produce the 
amount r J- in a unit of time at the rate r ? This is easily determin- 
ed, because if $1 is produced in a unit of time by an amount r +-> the 
amount r +~ will be produced by a proportional sum. From which 
we have the following : 1 : r +~i 11 r+~? 5 (t+^Y* We have before 
seen that r +- will, in a unit of time, at a rate of interest r, produce 
$1 ; therefore (r+-) 2 is the amount that will, if invested at compound 
interest, at the rate r, for two units of time, produce $1. 

In like manner, it may be shown that the expression for the 
amount that will produce $1 in three units of time is ( r |-) 3 ; and 
the rule is general : First divide unity by unity plus the rate of in- 
terest^ and then raise this quantity to a power, the exponent of which 
is the number of units of time.'' 1 

In illustration, suppose that the time is three months, the rate of 
interest is 1 per cent per month, interest compounded monthly, and 
the amount to be paid is $1000. In the first place, make the calcu- 
lation on the supposition that the amount to be paid at the end 
of three months is $1. The expression then becomes ( T .Jy) 3 = 



NOTES ON LIFE INSURANCE. 69 

$0.97059015. Multiply this by 1000, and we have $970.59, which is 
the amount that will produce $1000 in three months, at compound 
interest, at the rate of one per cent per month. 

In further illustration, suppose the question is, how much money 
will produce $1000 in fifty years, if invested at compound interest, 
at 4 per cent per annum ? First, calculate the amount that will pro- 
duce $1. The expression in this case becomes ( T .Jj) 60 . Divide 1 by 
1.04, the result is 0.96153846 ; multiply this by itself forty-nine 
times, or else find the logarithm corresponding to this number, mul- 
tiply it by 50, and find from the table the number corresponding to 
this logarithm. By either process, the result is $0.14071262. This 
is the amount that will produce $1 in fifty years, if invested at 4 
per cent per annum, compound interest ; multiply it by 1000, and 
we have $140.71, which is the amount that will produce $1000 in 
fifty years, at 4 per cent compound interest. 

In further illustration of the subject of compound interest, sup- 
pose we represent by v the amount that will, if invested at a rate of 
interest r, per annum, produce $1 in one year. Then r times v di- 
vided by 100 will represent the interest on v at the rate r for one 

year — that is, — - is the interest. Add this to the principal, which 

is v, and the two together must, from the condition imposed, be 

equal to one dollar. Therefore, we have the equation v ~|- — - =$1. 

Multiply both members of this equation by 100, in order to clear it 
of fractions, and we have 100 v-\-rv = 100, or v (100 +r) = 100; hence 
100 



100 + r 

To find the amount that will produce $1 in two years, the rate of 
interest per annum being represented by r, and the interest com- 
pounded yearly. Designate this amount by vJ" 

Now, if we multiply v" by the rate of interest r, and divide the 

product by 100, we will obtain an expression which represents the 

interest on v" at the rate r for the first year. The interest for the 

rv" 
first year is therefore represented by — . Add this interest to the 

r v" 
principal v" and we have v" -f- — , which is the sum to be placed 

at interest at the beginning of the second year. This sum v"-\- 

rv" 

— - , multiplied by r, and the product divided by 100, will give us 



70 NOTES ON LIFE INSURANCE. 

— («"+-— J, which is the interest during the second year. The 

original sum v", with the interest for the first year and the interest 
for the second year added to it, is equal to one dollar ; therefore we 

have v"-^~\- f^o( w "+^j=^ 1 - Multiplying both members of 

this equation by 10,000, in order to clear it of denominators, Ave 
have 10,000v" + 200™" -f- r 1 v" — 10,000, or ** (10,000 -f- 200r 

+r") = 10, 000. Hence, v'= ^°_2 _ 

' ; ' ' 10,000 + 2007- + r 3 • 

The algebraic expression for v, in terms of r, namely, v = 
will, when multiplied by itself, or raised to the second pow- 



100-fr 

, , 10,000 , ■ • . •._ 

er, become v =— — — — — ; and this is the precise expression 

' 10,000 + 200;- + ^ ' F ^ 

found above for the value of v". Therefore, d'—v*, that is to say, 

the present value of one dollar, payable certain at the end of two 

years, at any rate of interest r, compounded annually, is equal to 

the present value of one dollar, payable certain at the end of one 

year, at the same rate of interest, raised to the second power. 

Calling the present value of one dollar, to be paid certain at the 
end of three years, v"\ and placing this at a rate of interest r, we 
find in a similar manner an algebraic expression for the value of 
v"'\ Having found this value, an inspection of the algebraic ex- 
pression will show that v'" is equal to v raised to the third power. 

In all cases the present value of one dollar (computed at any given 
rate of interest), to be paid certain at the end of one year, will, 
when raised to a power, the exponent of which is n, be equal to the 
present value of one dollar, to .be paid certain at the end of n years, 
interest being compounded annually. 

Notatiox. — Let I = the number of persons living at any age ac- 
cording to the mortality table used ; then 1 30 = number of persons 
living at age 30 ; and in general, l z = number living at any designa- 
ted age, x. 

Let n = the number of years that a policy has been in force. Then 
4+„ = the number of persons, as shown by the table, living n years 
after the policy was taken out at age x. Suppose the policy was 
taken out at age 40 and had been in force 10 years, then l x+n becomes 

Let 'd = number of deaths during any year, and d z = number of 
deaths in the year between age x and age ic + 1. If the age is 40, d, 
becomes d 40 , which represents the number of deaths given in the 



NOTES ON LIFE INSURANCE. 71 

table opposite age 40, which is the number of deaths between age 40 
• and age 41 ; d x+n — the number of deaths during the year between 
age x + n and age x + n + 1. Suppose the policy was taken at age 30 
and had been in force 6 years, then x=30, n=6, and d x+n =d 30+6 =d 36 , 
and is given in the table opposite age 36, and is the number of 
deaths between age 3G and age 37. 

Let v = the value in hand, or amount of money that will at a given 
rate of interest per annum produce $1 in one year. 

Net Single Premium.. — The foregoing notation being understood, 
we are ready to write out the expression for the value of the net 
single premium that will, at any age, insure $1 to be paid to the 
heirs of the insured at the end of the year in which he may die. 

The amount that will insure %\ the first year is equal to the amount 
that will produce $1 in one year multiplied by the fraction which 
represents, at the age the policy is issued, the chance that the in- 
sured will die during the first year, v is the amount that will, when 

increased by interest, produce $1 certain in one year ; — is the frac- 

tion which at age x represents the chance that the insured will die 
before he reaches age x 4-1. Therefore the amount that will at age 

x insure $1 for the first year is expressed by v X y? 

The amount that will at compound interest produce $1 in two 
years is v* ; the fraction which at age x represents the chance that 

the insured will die during the second year is -|~;* the amount 

that will at age x insure $1 for the second year is expressed by 
d ± 
k 
year is v z X-^« In like manner we obtain the amount that will if 

paid in hand at age x insure $1 for each separate year to the table 
limit. Add together all these respective yearly values, call the sum 
sP x , which is the symbol we will use to represent the net single pre- 
mium that will at age x insure $1 for whole life, and we have — 

* The fraction which at the beginning of the second year represents the chance that 

the insured will die during that year is expressed by ^fi^i ; but in order that there may be at 

h + i 
age x any chance that the insured will die in the second year, he must be alive at the beginning 

of that year. The chance at age x of the insured living to age cc+1 is expressed by --tJ. 

la 

Therefore -i±Ix-l±l=^±i=the fraction which at age x represents the chance that the 

Ix + 1 Ix Ix 

insured will die between age x+1 and age x+2. 



v * y^ _x+i^ rp^ amoun £ ^hat -win at age x insure $1 for the third 



72 NOTES ON" LIFE INSURANCE. 

sF x =v^+v'^l+v"^ + , etc., to the table limit; 

f>z f>x t x 

-p vd x + v*d x+l + v*d x+<i + , etc., to the table limit. 

or, si x _ — 

Multiplying both numerator and denominator of the second member 
of this equation by v% and we have — 

p v x+1 d x +v x+ *d x+l +v x+3 d x+ < i + , etc., to the table limit. 

Call the sum of the terms of the numerator of the second member of 
the equation M,. , and the denominator D x , and the equation becomes 

sP x =^. In like manner, S P S+1 =^£±1, s P a+2 ^[?±2, and sP x 



— — .w — ~ — ~^, *,* x+1 =- , — x+:e _ , - — x+n 

x •L'xn Mr+2 



M x+1 



M J+W 

Vx + n' 

The numerical values of M and D at each age have been calculated 
and placed in tables, as previously explained. 

The value at age x of the net single premium that will, if paid at 

age x + 1, insure $1 from that age to the table limit is, vX 

* + - ; because D x+1 ==v x+1 l x+l . The fraction which at age x repre- 
sents the chance that the insured will be alive at age x + 1, is ex- 
pressed by ^p. ; therefore the value in hand at age x of the net sin- 
gle premium that will effect the insurance of $1 for whole life, 
commencing at age x + 1, is expressed by —^r^- X -y^; which 

M M 

may be written ——==-— ^±*. The same reasoning will apply 

to age x + 2, age x + 3, and to any age, x + n. Therefore the amount 
that will if paid at age x insure $1 at age x+n and continue the in- 

M 

surance from that time for whole life is expressed by -—?• 

Value at age x of a life series of payments of $1 each, the first 
being paid in hand and one at the beginning of each year during 
life. — The first payment is $1. The amount that will, if paid in hand 
and increased by interest, produce $1 in one year is v. The fraction 
which at age x represents the chance that the person will be alive at 

age x+ 1 is ^±1 ; therefore the value at age x of the payment that is 

Vm 



NOTES ON LIFE INSURANCE. 73 

to be made at the beginning of the second year is v-^. In like 
manner the value at age x of the payment to be made at the begin- 
ning of the third year is v' 2 —. For the fourth year it is v 3 ^^. And 
so on to the table limit of life. Add together all these yearly values, 

call their sum A,, and we have the equation: A J p=$l + — ^4-^-^ + , 

l x l x 

etc., to table limit. Or, 

A 4 + ^4+i + ^ 2 4+2 + » etc., to table limit. 

Al __ — . x 

Multiplying both numerator and denominator of the second mem- 
ber of this equation by v s , and we have — 

a v% + v x+1 Z I+l + v x+2 4 +2 + , etc., to table limit. 

x v% 

We have previously represented v% by D x ; call the sum of the 
terms of the numerator of the second member of the equation N a , 

N 
and we have A x =* 

Net Annual Premium that icitt at age x insure $1 for whole life. — 
We have found that the net single premium that will at age x in- 

sure $1 for whole life is =r^. The value at the same age of a whole life 

N 
series of annual payments of $1 each is =-^. Therefore calling the net 

annual premium that will at age x insure %\ for whole life aP x , we 

N M * M 

have the proportion, =p : fp :: ■ $* : a ^*> from which aP x = ~^. 

If the insurance is for n years only, the expression for the net sin- 
gle premium becomes, x =r x+n ; because ^~ will effect the insu- 
lar 
ranee for whole life ; and —^ will, if paid in hand at age x, insure 

$1 from age x + n to the table limit. Therefore, the difference be- 
tween the two is the net single premium that will effect the insur- 
ance during the first n years. The net annual premium for n years 
that will effect the insurance of $1 for this term is expressed by 
M-M x+ r 
N,— N x+n 



; because the value at age x of n annual premiums of $1 



each, to be paid if the insured is alive at the time, is expressed by 

N n 

■ x x+rt ; and since we have previously found the net single pre- 



74 NOTES ON LIFE INSURANCE. 



■ ■ M M 

mium that will at age x insure $1 for n years is * — — , we have : 

-La,. 

X-N.+. . M.— ?C. ;. ; M,-M, + , 

The net annual premium for n years that will at age x insure $1 

M M 

for whole life is -^ — ^= — ; because the net single premium is -—, 

-^ x -^ x+n *J X 



and the value at age x of n annual payments of $1, each, to be paid if 
the insured is alive, is x * — . We therefore have the proportion: 
N-*W. M, ■ m . tn . M x 



V x my B a r ' N x — N 



x+n 



To insure at age x an Endowment of%\ to be paid to the insured 
at age x + n, in case he is alive at the latter age. — The amount that 
will if invested at age x produce $1 in n years is v n . The fraction 
that at age x represents the chance that the insured will be alive at age 

x + n is -~; therefore the amount that will at age x insure $1, to be 
l x 

v n l 
paid to the insured in case he is alive at age x+n, is x+n . Multi- 

lx 

v x+n l 
tiplying both terms of this fraction by v% it becomes — -—■ ; but 

v x+ %+ n = 3\+«, therefore -~^ is the amount that will if paid at age 

x effect the endowment of $1 at age x + n. The net annual premi- 
um for n years that will be the equivalent of this net single premi- 
um is =^= — —^ — . This is obtained from the proportion: 

■N * -N x+n 

■£*x -^z+m . "x+n .. <J>i . J-* x+n 



T> x ' D x v- /N a -N, + ; 

Term Insurance combined with Endowment at the end of the Insur- 
ance. — There being in this case two different contracts, the simpler 
if not the shorter method will be to find as above the net premium 
for each, and then add the two premiums together. 

M 

The Columns JR and S. ^~- will at age x insure $1 for whole life ; 

M 

——■ will at age x insure $1, commencing at age x + 1, and then con- 
■*^* 

M 
tinue the insurance for whole life ; and in general -=~ will at age 



NOTES ON LIFE INSURANCE. 75 

x insure $1, commencing at agex + ?i, and continue the insurance for 
whole life. 

The quantity in the column headed R opposite any age, x, is equal 
to the sum arising from adding to M at that age all the Ms at old- 
er ages. The sum of these Ms divided by D x gives the net single 
premium that will at age x insure $1 the first year, $2 the second, 

$3 the third, and so on, increasing the insurance $1 each year to the 

■p 
table limit, and is expressed by =*. 

The amount that will, if paid in hand at age x, insure $1, com- 
mencing at age x + n, $2 at age x + n-hl, $3 at age cc-f-rc + 2, and so 

on, increasing the insurance $1 each year, is expressed by -~^ . 

Therefore the amount that will at age x insure $1 the first year, $2 
the second, $3 the third, and so on for n years, and then conti- 

nue to insure n times $1 to the table limit, is expressed by ~ ~~ 



The amount that will at age x insure $1, commencing at age x + n, 
and then continue the insurance for whole life, has been previously 

shown to be -z^~ ; therefore, the amount that will at age x insure 

n times %\ at age x + n, and continue the insurance for whole life, is 

nyc M 
expressed by — x+n . From which it is seen that the amount 

that will, if paid at age x, insure $1 the first year, $2 the second, 
83 the third, and so on, increasing the amount insured $1 each 
year for n years, and then ceasing to insure, is expressed by 

R,— R x+n — nM x+n 



D* 

The S column of the commutation tables is formed by adding to 
N at any age all the Ns of greater ages ; and as N at any age divid- 
ed by T> at the same age is the amount at that age that is equiva- 
lent to a life series of annual payments of $1 each ; S at any age, 
divided by D at the same age, is the amount at that age that is 
equivalent to a life series of annual payments of $1 the first year, 
$2 the second, $3 the third, and so on, increasing the annual pay- 
ment by $1 each successive year. 

The relation between sPx, Ax, and aPx. — The general expression 
for the amount that will, if paid at age x, insure $1 for whole life, is 



76 NOTES ON LIFE INSURANCE. 

(see page 72) , Pa = ^'* + ^^« + ^y etc -> tQ ta ^ ^. 

Noting that d x =l—l x+l ; d*+i=4+i— 4 +2 ; d x+ r=i x+i —l x+z , etc., the 
above expression may be written : 

p = ^ +1 (I— l +l )+v x+ * (l x+l —l x+ *)+v* +3 (4 +2 — 4 +3 ) + etc. 

v% 
By separating the negative from the positive portion of each term, 
writing v as a common factor of all the positive terms, and placing 
all the negative terms in parentheses, with minus sign prefixed, we 
have : 

cPx V ( vHx + V * +Hx+l + V * +Hx+ * + etC ^ ( V * +H *+* + v * +il *+> + vX+H *+* + etc - ) 

Referring to the general expression for A,, (page 73) it will be seen 

that the first term of the second member of the foregoing equation 

may be written vAx y and the second term will be identical with Ax, 

v x l 
provided we prefix to it the expression — - = 1 ; this term may, 

therefore, be written, A x — 1. From which we have sP x = wA x — (A s 
- 1) = vA x -A,+l=l-A x + vA ,=l-(A,-vA,)=l-(l-i>) A,= 

The net annual premium is obtained from the proportion, A x : 1 — 
(1— v)A x ::U : dP x . From which we have aP x =-r (1— v)=— — 

A x -N x 

(1-v). 

Net Single Return Premium, — To determine the amount that will, 
if paid in hand, insure %\ for whole life, to be paid at the end of any 
year in which the insured may die, and at the same time return the 
premium without interest, designate this net single premium by Z x . 

The amount insured for whole life is therefore $1 + Z x . Since —■ is the 

net single premium that will insure $1, the net single premium that will 

insure $1 + Z X is expressed by (1 + Z*)^. Therefore we have Z x = ^~ 

+ Z X ^; or, Z X D X =M X + Z X M X , or Z^-MJ-M,, or Z x = g-^^. 

Net Annual Return Premium. — In case the premiums on the re- 
turn plan are paid annually, we will designate the net annual pre- 
mium by Z' x . In this case the amount insured the first year is $1 -f 
Z' x ; the second year it is $1 + 2Z' X ; the third, $1 + 3Z' X , and so on, 
increasing each year the amount insured by one net annual premium. 



I 



NOTES ON LIFE INSURANCE. 77 

The amount in hand that will insure $1 the first year, $2 the se- 
cond, $3 the third, and so on, is expressed by =A therefore the 

amount that will insure Z' x the first year, 2Z' X the second, 3Z' X the 

Z' R 

third, and so on, is expressed by * ' . This is the net single pre- 

mium that will insure the return of the net annual Z' x premiums. Add 
to this the net single premium, =-±, that will insure $1, and we have -^~ 

Z' R 

+ * J , which expresses the net single premium that will insure $1 

and return the net annual premiums. But the value in hand of this 

series of annual Z' x premiums is Z' x ~, therefore we have the equa- 

TsT M "R M 

tion : Z' ~* = £ + Z' x £?, or Z' x (N -R x ) = M„ or Z' x = 



D X ~D X l -D m 9 ~ ~ * v * ~" " ~* ~ N— R; 
This is the net annual premium that will insure $1 and the return of 
all the net annual premiums at the end of the year in which the in- 
sured dies. 

Suppose the amount to be returned is the net annual premiums 
plus m per cent thereof. Designate the net annual pr^miumjin thfe 
case by Z" x . The amount insured the first year will be $1 + 
(1+m) Z"„ the second $1 + 2 (1+m) Z" x , the third $1 + 3 
(1 + m) Z" xy and so on. From this we form the equation : 

Z"- X ^=^+ (!+»)#'. X 5i orZ" a 



When Payments of Annual Premiums continue for n years only. 
— Call the net annual premium in this case Z'" x . The net single 
premium that will at age x insure $1 and return all the Z'" x net 
annual premiums that are paid in n years is expressed by 

=^- + Z'" x x J -p. z+n » But the annual premium Z'" x for n years is 

]sj- ]ST , 

equivalent to Z'" x X — * ^ J+ " paid in hand ; therefore we nave the 

X "N" M R R 

equation, Z'" x X — p x+ - = ^y+ %"* X ' p * + -. Multiplying by 

the common denominator, and transposing, we have : 

Z"'. | (N — X x+n )— (R — K x+n ) j. =M a , from which 

M 

T7III 



(N-N x+ „)-(R x -R 3+ J 



78 NOTES ON LIFE INSURANCE. 

To determine the net annual premium for n years that will pro- 
vide for the return of the premium and m per cent thereof in addi- 
tion. Call this net premium Z'V The amount insured the first 
year is % i + 1 + m Z'%, the second $1 + 2 X l+m Z'%, the third $1 + 
3 X 1 +m Z iv ^ and so on for n years. 

In this case we find : 



* (N x — X x+n )— T& (R — R 2+n ) 

This is the amount that will, if paid annually for n years, insure 
$1 for life, and return all the net premiums, with m per cent thereof 
in addition. 

Decreasing Net Annual Premiums to Insure $1 for Life. — First 
find the value in hand of a life series of annual premiums, the first of 
which is $1, and each succeeding premium is m per cent less than 
the one that precedes it. The first payment being $1, its value in 

hand may be represented by -j-. The second payment is $1 — 

m 100 — m „._ , ... 

— — = — . lne amount that will, at a rate of interest repre- 
sented by r produce $1 in one year is expressed by v ; therefore the 
amount that will, at the same rate of interest, produce — of a 

dollar is expressed by ■ — X v. But the second payment is 

only to be made in case the person is alive at the beginning of the 
second year ; therefore the value in hand of the second payment is 

expressed by X v X -y 1 ^ The third payment, to be made 

at the beginning of the third year, in case the person is alive, is m 
per cent less than the second. Its amount is then expressed by 

100 — m m /100 — m\ 10000— 100m — 100m + m 2 _ 

100 100 [ 100 ~J~~ 10000 _ 

10000— 200m + m\ / 100— m \ 2 
10000 _ 1 100 J ' 

Therefore the value in hand of the third payment of this decreas- 
ing series is expressed by ( ~ -l X v'X-^ 5 . Taking m per cent 

\ 100 / i x 

from ( ^— ] , we find the amount of the fourth payment to be 



NOTES ON LIFE INSURANCE. 79 

( J ; therefore the value in hand of this payment is 

[ — — ) X v* X 4ft In like manner the amount of each succeeding 

yearly payment is obtained. Designating Xv by the sym- 
bol v\ adding together the respective values in hand of the several 
decreasing annual premiums, and multiplying both numerator and 
denominator of the fraction by v 1 raised to the x power, we shall 
have an expression for the value in hand of a life series of decreas- 
ing annual premiums, each of which is m per cent less than the next 
preceding. Representing the numerator by N 1 , and the denomina- 
tor by D 1 , we have the equation : 

g, = ». w+ v, + , + r ^ + ^4 + , + et to tab]e lim . t 

Before calculating the D 1 and N 1 columns of the commutation 
table for decreasing premiums, the rate per cent of decrease must be 
designated. A set of D 1 and N 1 columns will have to be computed 
for each different rate of decrease. Suppose, for instance, the rate 
of decrease is ten per cent; the first payment being $1, the second 
$0.90, the third $0.81, and so on, making each payment ten per cent 
less than the next preceding ; and so for any other designated 
rate of decrease. 

Having determined the rate per cent of decrease and constructed 
the corresponding D 1 and N 1 columns, represent the value of the 
first net annual premium of this decreasing series by the symbol p x . 

Then since =— represents the value, at age x, of a life series of an- 

-*-' X 

nual premiums, the first of which is $1, and each succeeding one a 

certain per cent less than that which immediately precedes it, the 

value in hand, at age x, of a similar series of decreasing premiums, 

N 1 
the first of which is p m is expressed by p x X y^ 5 . But the quantity^ 

must be such that the value, at age x, of the life series of decreasing 
annual premiums will insure $1 for life. The net single premium 
that will, at age x, insure $1 for whole life, as previously shown, is 

:=A Therefore we have the equation : 

"a 

N 1 M MD 1 M D 1 

p a Xjjr=jf, «>-= j^r = p^ X ^r x = the first net annual pre- 
mium of this decreasing series for whole life. 



80 NOTES ON LIFE INSUKANCE. 

If the decreasing annual premiums are to be paid for n years 
only, call the first of this series p\ Then, since the value in hand of 
a life series of similar decreasing premiums for n years, the first of 

which is $1, is expressed by — * L jy~~ *~> the value of this series, the first 

N 1 — W 
term of which is p 1 ^ will be expressed by p x x X — ^-yt — — . But the 

value in hand of this series of n annual decreasing premiums must, 

M 
in order to effect the insurance of $1, be equal to y— . Therefore, the 

equation : 

ftp x 1 M M D 1 M D 1 

F*a pn — T . , or,p x — xfl . AJ - 1 vfT— p, A 



B\ — D, ' w >? * (NV-N'.+.JD. D. " N 1 .— NU 



JOINT LIVES. 

Note.—" The probability of the happening of any event is to be understood as the ratio of 
the chances by which the event may happen, to all the chances by which it may either happen 
or fail ; and it may be expressed by .a fraction whose numerator is the number of chances 
whereby the event may happen, and whose denominator is the number of chances whereby it 
may either happen or fail. Thus, if there be a chances for the happening of an event, and b 
chances for it not happening, then will the probability of such an event taking place be repre- 
sented by — -r-j~ In like manner, the probability of any event failing (or of not happening) 

may be expressed by a fraction whose numerator is the number of chances whereby it may fail, 
and whose denominator is, as before, the whole number of chances whereby it may either hap- 
pen or fail. Thus, the probability of the above event failing will be truly expressed by —_r-r' 

Since the sum of the two fractions, representing the probabilities of the happening and of the 
failing of any event, is equal to unity, it follows that, one of them being given, the other may 

be found by subtraction. Thus, the probability of an event happening being denoted by -, 

the probability of the same event failing will be truly represented by 1 -— r = ., 

a + o a-\-b 
and vice versa. The probability of the happening of several events that are independent of 
each other is equal to the product of the probabilities of the happening of each event con- 
sidered separately ; and the probability of the failing of any number of independent events is 
equal to the product of the probability of the failing of each event considered separately."— 
{Doctrine of Chances.) 

Suppose that the two joint lives are aged respectively x and y. 
Then the chance, at the time the contract is made, that the person 
aged x will live till he is aged x + 1 is expressed by the fraction 

-y-, and the value in hand of $1, payable at the beginning of the 

second year on condition that the person is alive at the age x + 1, 

will be v-yL. The chance at the time the contract is made that the 

person aged y will live to age y + 1 is expressed by ^~. Therefore 



NOTES ON LIFE INSURANCE. 81 



-y-Xy- is the fraction that expresses the probability, at the time 

4 4 

the contract is made, that both the joint lives will be in existence at 
the beginning of the second year. Consequently 1 — y^X-y^ is 

4 4 

the expression which represents, at the time the contract is made, 
the probability that both of the joint lives will not be in existence 
at the beginning of the second year. In other words, it expresses 
the chance that either one or the other of the two joint lives may 

die during the first year. Therefore v X (l y^X-y^l will insure 

81 on these two joint lives the first year. The expression may be 

The fraction which represents at the beginning of the second year 
the probability at that time that these two joint lives will continue 

in being together during that year is expressed by f^ X y^, and 

4+i 4+i 
the probability at the beginning of the second year that the two 
lives will not continue in being together during the second year is 

expressed by 1 — l f^ X ^. 
4+i 4+i 
At the beginning of the first year, there can be no chance that 
the insurance will become due at the end of the second \ear, unless 
both of the insured persons live until the beginning of that year ; 
therefore we must' in the previous expression impose this additional 
condition. The expression which represents, at the time the con- 
tract is made, the chance that the two lives will continue in being 

until the beginning of the second year is ~- X y^* There- 

4 4 

4+2 s. 4+A X x 4+i 4+1 



fore 1-pXy 1 X-y 2 X -y 2 expresses the chance at the time 

V 4+i 4+i' 4 4 
the contract is made that either one or the other of the two joint 
lives will fail during the second year. The above expression may 

be written (kl^_ki±?W ^^±1. B y striking out the factor 

\4+i4+i 4+i4+i/ 44 
4+A+i, which is common to the numerator and denominator, it be- 
comes x+1 y+1 7 / J+2 y+2 - Multiply this by v 2 , and we have the amount 

44 
that will, if paid at age x y, insure $1 on these two joint lives 
during the second year. 

By a process entirely similar, we find for the third year, 



82 NOTES ON LIFE INSURANCE. 

v* * +g y+2 x+3 y+3 , and in like manner for each successive year to 

f'J'y 

the limit of the mortality table. Add together all these respective 
yearly values, and we have the net single premium that will at age 
x y insure these two joint lives for whole life. 

Representing this net single premium by sP xy , we have sP xy == 

v u~ + v IJ, + " u, + etc - 

to the table limit. 

By separating the negative from the positive terms and writing 
v as the common factor of all the positive terms, the equation be- 
comes : 

sr xy — v =-= 

My 

(vh+Jt+i + v%+J ¥ +* + v%+ 3 l y+s + etc.) 

Uy 

Multiply both numerator and denominator of these fractions by v x , 
and we have : 

e -p _ JvrlJ, + v x+ % + J y+ i + v x+ % + J y+ * + etc.) 

fo* +1 W y+ i + v* + % + J y+ > + v x+ % + J y+ * + etc.) 

V'ljy 

Represent the second factor of the first term of the second member 
of this equation by =p, and the equation becomes sP xy = v=~ — 

"xy "xy 

fe- 1 )=^-^- 1 >' 0r ^ =1+ ''S-S =1+ S 

N B (1 — v ) "S 

(v — 1)=1— (1— «)^2 = ^ l ' * y ; this is equal to 

1 — (1— v)^. 

The net annual premium that will insure $1 on the two joint lives 
is obtained by the proportion : 
N^ . P„ — (1— ^)N xy . P zy — (1— ^)N ay _ D, y 

6~ : s: • • n * s; n;" (1 ~ v) 

= _L_ (1 _ V) . 

When a third life aged z is associated in joint insurance with the 
two lives aged x and y, the fraction which expresses the chance that 
the three lives will continue in being together during the first year 

of the insurance is x+1 /t) ' +1 . And the chance at the time the in- 



NOTES ON LIFE INSURANCE. 83 

surance is effected that the three lives will not continue in being 

together during the first year is 1 * H **' * +1 . The amount that 

will if paid at the time the policy is issued insure these three joint 
lives for one year is v 1 1 * +1 //~/ j* ^ ne principles already ex- 
plained apply to any number of joint lives. 

The formulas used in calculating net values in the insurance of 
joint lives are similar to those for single lives, using, however, joint- 
life commutation columns in place of corresponding columns for 
single lives. 



84 NOTES ON LIFE INSURANCE. 



CHAPTER YII. 

THE DEPOSIT, USUALLY CALLED RESERVE. 

Full-paid Insurance. — In case a policy has been fully paid for, the 
deposit that must be held by the insurer to the credit of the 
policy is the net amount in hand that will at that time effect the 
insurance for the unexpired term of the policy ; this amount is com- 
puted on the basis of a designated table of mortality and rate of in- 
terest, both of which are, in most States, specified by law, and form 
the standard of legal net values in life insurance. The net single 
premium at age cc, that will insure $1 for whole life, the value of 

M 
which is expressed by -^r > having been paid at that age, the law 

requires that the insurer, in whose hands these trust funds have been 

placed, shall have in possession to the credit of the insured, at the 

time the policy-holder has reached the age x + n, an amount equal 

M 
to yt~ ' ^ ecause this is the net single premium that will at age 

x + n effect the insurance on the designated legal data. 

In general, whatever may be the kind of policy, an original net 
single premium paid at age x is the amount that will, on the desig- 
nated data, pay for insurance each successive year, and leave in the 
hands of the company, at the end of each year, an amount sufficient 
to effect at that time all the insurance still called for by the terms 
of the contract. 

Whole-Life Insurance, by Net Annual Premiums. — In case a whole- 
life policy is paid for by net annual premiums, the yearly payments 
being equal to each other ; we have previously seen that the amount 
of the annual premium is determined by the condition imposed that 
the value in hand of the life series of annual premiums shall be equal 

M 

to the net single premium. The latter at age x is equal to -^. 

At the same age the equivalent net annual premium is -~. The 
value in hand at age x of a life series of annual payments of $1 each 
is expressed by ^r ; therefore the value at the same age of a life 



NOTES ON LIFE INSURANCE. 85 

M x . M 

series of annual payments each equal to -^~ is expressed by -=^ X 

—^ = —^ ; which satisfies the conditions imposed — namely, that the 

value in hand at age xoi the life series of net annual premiums shall 
be equal to the net single premium that will on the designated data 
effect the insurance at age x. 

At age x + n> the net single premium that will insure $1 for whole 

life is pr^. The net annual premium that will at age x + n effect 

the insurance is x+n . But at age x + n the insured does not pay 

M 

the net annual premium ^=^ due to this age, but has only to pay 

M x 

the net annual premium ^— due to the age x, which is less than 

that due to age x + n. 

The difference in value at age x + n between a life series of annual 

payments each equal to y^ and a similar series each equal to -~ 

is by law required to be held by the insurer to the credit of the 
policy at age x + n. This amount may be calculated either by sub- 
tracting from the net single premium at age x+n the value in hand 
at that age of a life series of net annual premiums each equal to 

M s . . i . M x+n M x N x V „ . 

-^-, which gives ^-^ ^-X y% 5 or "Y subtracting the net 

M x M x 

annual premium -— from the net annual premium a+re , and then 

™x ^x+n 

finding the value at age x + n of a life series of annual premiums 

M x+n M x __^ _._ /M x+n _M x \ ■ K +M 



each equal to ^±? — ^, which gives (jf^ — ^f ) X^, and 

ML" N* M x N x+n X n M x+n " M x * k+« 

may be written ^ X ^ -~ X ^ = jf* — ~ X ^ 

^x+n Ux+n ^x ^x+n ^ x+n ^x ^ x+n 

In still further illustration of the method of computing the 
amount that must be held in deposit at the end of any year for a 
whole-life policy of $1 paid for by equal net annual premiums; 
take the expressions previously obtained for the net single pre- 
mium in terms of A x and of N x and D x , namely, sP x =l — (1 — v) A x = 

N' 
1 — (1 — v) ~j~ ; and that for the net annual premium, namely, aP x = 

— (1 — v) = -^ — (1 — v). At age x + n these formulas become 



86 NOTES ON LIFE INSURANCE. 

'P.+. = l-(l-») A x+n = l-(l-u) 5a- a nd aP x+n = -i 

(l_)=^-(l_ ). 

The difference between the net annual premium at age x and that 
at age x + n is dP x+n , — aP x . The value at age x + n of a series of 
annual premiums, each equal to this difference, is obtained from the 
proportion, $1 : (aP x+n — aP x ) : : A x+n : A, +n (aY x+n — aP x ). 

The fourth term of this proportion gives the deposit or " reserve " 
for this policy at age x+n. 

Another expression for the deposit is obtained by taking the net 
single premium, at age x + n, and subtracting from it the value at 
that age of the future life series of dP x annual premiums. This is 
the amount that must be on hand in deposit, to cause the future 
dP x annual premiums and the deposit to be equivalent to the fu- 
ture aP x+n annual premiums. From this we have the equation, 
sP x + n — {a-P* X Ah-») — tne d&posit at the end of n years from the 
date of the policy. Substituting for sP x ^ n its value, 1 — (1 — v) A x+n , 

and for aP x its value, -r (1 — u), and we have 1 — (1 — v) A x _ i _ n — 

-r (1 — v) \A*+» == the deposit at the end of n years. And as the 

two expressions (1 — v) A x+n have different signs, they cancel each 
other, and we have 1 — -y 1 ^ = " deposit " at the end of n years. 

Joint Lives. — In case the X and D columns for joint lives are con- 
structed and the M column is not, the deposit is calculated by the 
formulas involving the terms N a and D x , or A^. For two joint lives 

A 

paid for by net annual premiums, the formula is 1 *+»•*+ ; 4 

similar formula is used when there are more than two joint lives. 

Decreasing Premiums. — The first net annual premium of the de- 
creasing series as previously shown is expressed by =^- X ^f. The 

second premium is m per cent less than the first. The value of 
a series of decreasing annual premiums, the first of which is $1, and 
each premium thereafter is m per cent less than that immediately 
preceding, is expressed by A' x . The value at age cc + 1 of a series of 
regularly decreasing net annual premiums, the first of which is p — 
■fffi, is expressed by A' iM X(p — iTcr)- Therefore the deposit required 



NOTES ON LIFE INSURANCE. 87 

at the end of the first year is * +1 — A' x + , (p-* -jyb) ' Tlie premium 
at age x + 2 is ( jo-f/y ) a . The deposit at the end of the second year 
is therefore 1 ^ J + 8 A' a + 9 (jt? — fVy-) a . At age &-f w the expression 

for the deposit becomes * +n — A', + « (j9 — fVrr)* 

Note.— The quantity represented by A' in the above expression must not be based on a rate 
of interest greater than that designated as the basis of valuation . 

Annual Payments for a Years.— In case a whole-life policy for 
$1 is to be paid for by equal annual premiums in a years, each an- 

M 

nual premium is expressed by r-= =^ . The deposit at the end of 

n years, but before a years have elapsed, is, ^— z =^ r-j — X 

J + " t±^ t After a years have elapsed, the policy is full paid, 

and the deposit at the end of any year must be equal to the net single 
premium that will at that age effect the insurance. 

Note.— The general reader not specially interested in return premium policies is advised to 
pass at once to page 93. 

Return Premium. — The net single premium that will, at age x, 
insure $1 for whole life, and return this net single premium plus m 
M* 

■(l+m) 

tually paid, and which is to be returned, is m per cent greater than this 

net single premium, the amount insured will then be $1 -f 

(l+m)M a 



per cent thereof, is .=- T — - — r^=-. Assuming that the premium ac- 

v ' T> x — (l+m)M x & r 



D x — (l+ra)M x 



The net single premium that will, at age x + n, in- 



M 

sure $1 for whole life, is * + n , and the net single premium that 

■D* + n 

.„ . , • (l+m)M, . M x+ (l+m )M x 

will, at age x + n, insure ^ /n — X7 . r - is ^r X -r^ yz r^r- 

LV— (1 + m) M x D x + „ D— ( 1 + m) M x 

M 

Therefore the reserve is in this case expressed by T . x +n + 

"x + » 

M, + . (l + m)M, _ M, + (l+m)M, M, + 

D 1+ „ A D,— (l+m)M, D I+ „ AV ^D-(l+m)Mj D„ +n A 

D x -(l+m)M, ) ~ D, + „(D,-(l+m)M.) ~ dep ° Slt tOT 

this policy n years after issue. 



88 NOTES ON LIFE INSURANCE. 

The net annual premium that will insure $1 for whole life, and re- 
turn all the net annual premiums paid plus m per cent thereof, is ex- 

pressed by ^ — -, — - — 777-- Assuming that the annual premiums 

actually paid are each m per cent greater than this net, it will then 

become ^— — ; — - — ttt-* The annual premiums paid in n years 
N s — (l+m)R x r r 

amount to n X ^ — 7 -. \*-r» • The net single premium that will, 
N s — (l+m)R z r . 

M . 

at age x 4- ?i, insure this amount for whole life, is * X W X 

(l+m)M, 
N,— (l+ro)R, 

The net single premium that will, at age z + n, insure $1 the first 
year, $2 the second, 83 the third, and so on to the table limit, is 

* + " ; therefore the net single premium that will, at age x + ?i, in- 
sure the return of the annual premiums yet to be paid, is x+n X 



(l+f»)M, 



IW* 



N,— (l + w)R, 

The value in hand at age x + n of the net annual premiums yet to 

... . N I+n M x 

be paid is == x » 

Therefore we have the deposit at the end of n years expressed 
by: 

M z+n M J+ , (l + m)M, R^ (l + m)M, _K z+tt 

Dh-. ^U X N x -(l+m)B, + D^ l X N,~(l+m)R, D, +n 

X N X — (l + m)R, 

Noticing that f—^X n X xr v ,. * = ^ ,_ , p X^X 

D, +fl N,— (l + m)R, N s — (l+/w)R x 

and that ^ X J 1 ?*^ = XT , M * „ X ( 1+m > R ^ 



D* + » N,~(l m)R x ~ N,^(1+«*)E. " D x+n 

W ehave^-^ ^ ^ J ^ +n -(l + ^ (R z+w+ ,xM. +n ) | 
D,+n N.— (l+m)R,l T> z+n j 

reserve for this policy n years after issue. 



NOTES ON LIFE INSURANCE. 89 

The net single premium that will, at age x, insure $1, to be paid at 
age x + z, in case the insured is alive, and return this net premium 
plus m per cent thereof, if the insured dies before age x + z, is ex- 
pressed by :pr — t- ^t u — r- In case the insured pays m per 

r D x — (l + m)(M, — M x+l ) 

cent in addition to this net single premium, the premium actually paid, 

-ii-ii i -i • (l+ m )D:H-« 

and which has to be returned, is ^ 77 r-^nf — ™ — r 

D„— ( 1 + m) (M- M x+Z ) 

At age x + n the net single premium that will insure $1 until age 

x + z is expressed by M * + "~ M * +Z ; therefore Mj+ "~ M ' ±' X 

(l+m)D,+, _ IU (l+m)(M, + ,— M. + .) = , 

D B -(l+m) (M x -M x+Z ) ~ D x+n X D -(1 + m) (M a -M s+ ,) 

single premium that will, at age * + n, insure p ' L ™ / M ^ M y 

The net single premium that will, at age x-\-n, insure $1, to be 

paid to the insured, in case he is alive at age x -f- z, is * + z . From 

±> x + n 

the above, we have t— ± — r- -^—^ < ^ — x ,,, — ,., : > = 

D* + » r>,+n lL\-(l+m)(M s — M s+ ,) J 

9=±«ii4 (l+m)(M a + n -M a + 2 ) j._P, + . v 

D^r^.-ll+ffi) (M. — M. + I ) J ~L\ +n X 
( D,— (1 +m) (M - M, + ,) + (1 +m) (M, + . — M, + .) ) 
\ ~ D.-(l + m)(M.-M, + I ) [~ 

Di _ (1+m)(Mx -M I+2 ) X LV» -the deposit 

for this policy n years after issue. 

To obtain the net annual premium that will at age x insure $1, to 
be paid at age x + z in case the insured is alive, and if he dies before 
age x + z, the net annual premiums to be returned ; call this net an- 
nual premium W. Then since at age x the net single premium that 
will insure $1 the first year, $2 the second, $3 the third, and so on to 

the table limit, is expressed by =^, the net single premium that will 

at age x insure W the first year, 2 W the second year, 3W the third 

year, and so on to table limit, is expressed by — X W. 

The net single premium that will at age x insure W at age x + z, 
2W at age jc + s + 1, 3W at age x + z + 2, and so on to the table limit, 

is expressed by -^xW. 

±J x. 



90 NOTES ON LIFE INSURANCE. 

The net single premium that will at age x insure W the first year, 
2W the second, 3 W the third, and so on, increasing by W each year 
for z years, and then insure zxW for whole life, is expressed by 

* — -- x W. But the insurance of z X W from age x + z to the table 

limit is not included in the policy now under consideration, and it 

must therefore be deducted. The net single premium that will at 

age x insure $1 for life, to commence, however, from age x + z, is ex- 

M M 

pressed by -~^, Therefore ~X2XW is the net single premium 

that will at age x insure the amount zxW from age x -f z to table 
limit. Hence, the net single premium that will at .age x insure W 
the first year, 2W the second, 3W the third, and so on for z years, 
at which time this insurance ceases, is expressed by, 



R,-R I+ -2xM J+! 



xw. 



The net single premium that will at age x insure $1, to be paid at 
age x + z i£ the insured is then alive, is expressed by -^~- There- 
fore the net single premium that will in this case effect all the in- 
surance required is expressed by -ry- z + — f± ^ — xW. 

The value at age x of annual life premiums for z years each equal 

^[ N 

to "W is x ^ t+ ' xW. From this we have the equation : 
■*-'* 

Wx ^-N f±f = D itf + Wx R J -R I+ -zxM s+2 _ 



D x T> x D x 

W | (N— K x+Z )— (R — R I+2 — zxM x+z ) |=D I+r , or 

W = _— — — , * + * tt — r = the net annual premium 

(N— N x+2 ) — (R — R x+ — z X M x+Z ) r 

that will insure $1 to be paid at age x + z if the insured is then 
alive, and return all the net premiums paid in case the insured dies 
before that time. 

If the net annual premium is calculated to return itself plus m per 
cent thereof, the above expression is modified by introducing the 
factor (1 + m) in the last term of the denominator which involves H x , 
R i+Z , and sxM x+s . It will then become : 

D^ 

N, — N„ + , — (1+m) (R, — R, +! — sxM^,)' 



NOTES ON LIFE INSURANCE. 91 

The reserve on the above form of policy n years after issue is 
found as follows : 

The amount that will at age x + n insure ?iX W until age x + z is, 

~^r — -X»xW. 

Mr+n 

The amount that will at age x + n insure W the first year, 2\V 
the second, 3W the third, and so on until age x + z, is, 
j R x+n — R x+ , — (z — n) M, + , ) 

i iv» r 

The amount that will at age x + n insure an endowment of $1 at 

ao;e « + z is 



Therefore the net single premium that will at age x + n effect the 
required insurance and endowment is expressed by, 

,M, -M, + A + w / R, + ,-R (,-n)]>^ \ + D^ 

V -L'i+n / * ^*+n / Mc+» 

From this subtract the present value of the future net annual pre- 
miums, W, for a number of years. equal to z — n, which value is 

W — '—- — , and we have the expression for the reserve for this 

policy at age x + n : 

( n(M x+n —M x+z ) R J+n — R x+Z — (z — n)M x+t N x+n — K x+Z j 

( D, + n 7 V X+n D, + n f + 

Substituting for W its value, we have — 

D.+z j w x M J+W + Rx+n— Rx+ — g x M J+2 — X x+ „ + K J+ , | 
D x+W l N x -N x+z -(R x -R x+z -zM x+z ) r 01 

Dx+, j ^xM x+w +R I+ „-R x+ -gxM x+ -N x+w +N x+2 +N x -N x+ -(R -R x+ -gxM a+g ) i 
I) x+rt 1 N x -N x+2 -(R x -R x+z -z X M x+z ) f 

D x + Z ( N,— N x+n — (R— R x +W — nxMJ ) 

In case of premiums loaded m per cent, the expression is modified 
as follows : 

D*+ e j N x — N i+ - (1 x m ) (R-R x+n — n M x+n ) i 
T> x+n \ N-N x+z -(lxm)(R_R x+2 _zM x+z ) l 

To obtain the reserve that must be held at the end of n years to 



yS4 NOTES ON LIFE INSURANCE. 

the credit of a policy issued at age x to secure 81 at age x + z or at 
previous death : in either case, the actual premiums paid to be re- 
turned, together with the amount of insurance, the policy to be 
paid for by annual premiums. 

The net annual premium that will at age x insure $1 for z years, 

and an endowment of $1 at age x+z, is 



The net annual premium that will at age x insure $1 the first year, 
$2 the second, $3 the third, and so on for z years, and insure the 
endowment of $z at age x + z, is, 

n,—n x+z —z x m z+2 + z x T) x+Z 



N„ — N 



-2 



Represent the net annual premium, the value of which we wish to 
find in this case, by W'. We will then have — 



W'X 



R* — R«+, — g X M, + , + z X T> x+Z 



the net annual premium that will at age x insure W the first year, 
2W the second, 3W the third, and so on for z years, and insure the 
endowment of z X W at age x + z. 

%[ -^1 _I_;D 

Add this to the net annual premium * — ~= — — that will pay 

™ x -N x+z 

for the insurance and endowment of Si for z years, and we have — 

W/ _ M. — M^..+D^, w , R, — R, + , — gxM, + . + gxD, + . Qr 

N 2 — N x+Z N x — N x+ , ' 

W'-j p-^)-(R-K, + :-2XM I+! + 2 xD x+! ) ^ = M--M x+z+ jy x+ , 



(N-X x+! )-(R x -R x+s -zxM x+z +zxVx +s ) 



When it is desired that the net annual premium shall insure $1 as 
above, and provide for the return of the net annual premiums plus m 
per cent thereof ; call this net annual premium W 9 and we have, 

w , = M x — M x+Z + T> x+Z 

N, — N, +I — (1 + m) (R, — R, +z — zx M I+2 + z x T> x+Z )' 



The reserve at the end of n years is found as follows : 
The net single premium that will at age x + n insure $1 until age 
x + z, and an endowment of $1 at age x + z, is expressed by — 



NOTES ON LIFE INSURANCE. 93 

Suppose the annual premium actually paid is (1+m) W # . These 
annual premiums paid for n years will amount to wX(l+w).W; 
and at age x + n the net single premium that will insure nx 
(l+m)W until age x + z, and secure an endowment of the same 

amount at that age, is expressed by n X ( 1 + m)W X - x+ 



The net single premium that will at age x + n insure (l+m)W"the 
first year, 2 X (l+wi)W the second, and so on until age se + z, and se- 
cure an endowment of (z — n) x (1+m) W" at that age, is, 

(i +m ) "w" R * +n ~ Rj+ * ~ ^ ~ n ^ lx+z + ^ ~ n ^*+\ 

From the sum of the above three net single premiums subtract the 
expression which gives at age x + n the value of the net annual pre- 
miums to be paid between age x + n and age x + z, namely, 

-L'x+n 

and we have the required deposit. 



A GENERAL FORMULA FOR CALCULATING THE DEPOSIT OR RESERVE. 

The letter H is used to express this value. This symbol comes 
from the question, " How much must be in deposit ?" The expres- 
sion for the deposit at the end of the first policy year is H x+1 . 
The amount at risk during the first year will therefore be equal to 
81 — H x+1 . The amount that will at age x insure $1 for one 

year is v-j. The amount that will at the same age insure $1 — H a+1 

for one year is therefore v-f(l — H^) ; subtract this amount from 

the net premium which is represented by P x and we shall have, in 

case the insurance is paid for by a net single premium, sP x — v— x 

(1 — H I+] ). This is the expression for that part of the net premium 
that goes to form the deposit or reserve that must be held at the 
end of the first year. This part of the premium paid at the begin- 
ning of the year will, when increased by net interest, be the 
amount that must be in deposit at the end of the first year. 



94 NOTES ON LIFE INSURANCE. 

Let the ratio of interest be represented by r'. Observe that this 
r' is not the rate of interest ; it is a quantity which will, when the 
principal is multiplied by this quantity, produce what the principal 
will amount to when increased by net interest for one year. For 
instance, v being the principal and r' the ratio of interest, r'v is 
equal to one dollar ; and of course r' is equal to one divided by v. 

From this we have the equation, 

H, +1 =r'[«P-w|( 1 - H >+.)]- 

If the insurance is effected by the payment of annual premiums, 
each equal in amount and designated by dP z , the equation becomes, 

H, +1 =r' [«P _„|(l_H I+1 )]. 

This equation contains but one unknown quantity, namely, H x+1 . 
Performing the operations indicated in the second member, we 
have, 

H^r'a^-r'v^ + r'v^K^. 

Representing v y by c xi the equation becomes, 

K x+1 =r'aP x —r'c x + r'c ,H I+1 . 

Transposing the third term of the second member, we have, 

H z+1 — r'cM x+l =r'dP x — r'c x , and 
H z+1 (1— r'c x )—r'(aP x — c z \ or 

r' 

Call r— , w,, and we have 

1 — r c x 

K x+l =u x (aP x —c x ). 

Represent the amount that must be in deposit at the end of the 
second year by H z+2 . Then the amount at risk the second year will 
be expressed by $1 — H i+8 . The amount that will, if paid at the 

beginning of the second year, insure $1 during the year, is v -~. 

w 

Therefore v ^(1 — H I+2 ) is the amount that will, if paid at the be- 

ginning of the second year, insure the amount at risk daring that 
year. Subtract this from the net annual premium paid at the be- 
ginning of the second year, and that part of this premium which 



NOTES ON LIFE INSURANCE. 95 

remains will go to form deposit at the end of the second year. But 
the deposit at the end of the first year also goes to form deposit at 
the end of the second year. The amount of net funds on hand at 
the beginning of the second year, just after the second net annual 

premium is paid, is represented by H, +1 + aP x . Deduct v y^(l — H x+2 ), 

w 
which is that portion of the second net annual premium that will 
be required to effect insurance on the amount at risk during the 

second year, and we have H x+l + aP x — v P^(l — H, +2 ). This is 

the amount on hand at the beginning of the second year that goes 
to form deposit at the end of the year. Increase this by net in- 
terest for one year by multiplying the whole expression by /, and 
the result gives the amount that must be on deposit at the end of 
the second year ; hence we have the equation, 

H, + ,=r' ^H x+1 + «P-^(l-H x+2 )). 
Call v f^, e m+1 , and we have, 

*x+l 

H, + ,=rYH, +1 + aP,— c x+l (1— H X+2 )V or 
H, +2 =r' (H x+1 + aP x —c x+l + c x+l X H x+2 ). 

Transposing to the first member that term of the second member 
which contains the unknown quantity H x+2 , and we have, 

H x+2 — r'c x+1 X H x+2 = r' (K x+l + aP x — c x+1 ), or 
H x+2 (1 — r'c 9+1 ) = r' (H x+1 + aP x — c x+1 ), from which 

H x+2 = - I (H x+1 + aY x — c, +1 ). 

1 1 C x+\ 



T 

Call ; , u x+1 , and we have, 

J- T <?^_i_i 



r 

"'x+\ 

H x+2 = u x+l (H x+l + aP x — c x+1 ). 



From this equation, the amount that must be in deposit at the end 
of the second year can be obtained, provided we have previously 
calculated the deposit for the end of the first year. 

The amount in deposit at the end of the third year is H x+3 . The 
amount at risk during the year is therefore $1 — H x+3 . The amount 
that will, if paid at the beginning of the third year, insure $1 for 



96 NOTES ON LIFE INSURANCE. 

that year is vf^-, therefore v~ — (1 — H z+3 ) is the amount that will, 

if paid at the beginning of the third year, insure the amount at risk 
during that year. Subtract this from the net annual premium, and 
we obtain that part of the third premium that goes to produce de- 
posit at the end of the third year. Hence we have, 

H i+3 = r' [H x+2 + a-P — vf^ (1-H^ 3 )]. 
Call v y—^i <?*+2j and we have, 

4+2 

H x+3 = / (H z+2 + a? x — c x + 2 ) + r' c e+2 X H^g 

Transposing, we have, 

H z+ 3 — r' c x+2 X H r+3 = / (K x+i + <zP a — c*») 9 or 
H, +3 (1 — r' c z+2 ) = r (H^ 2 + aP„ — e^ or 

H x + 3 = = — -, (H z+2 + aP, — c^). 

l r c z _!_ 2 

Call : , U-. o, and we have 

H z _ 3 = *W» (H z+2 + aP, — c z+2 ). 

The deposit at the end of the third year can be calculated from 
this equation, after the deposit for the end of the second year has 
been found. 

In a manner entirely similar, we find, 

H,_4 = «*,+3 (H z _ 3 -r aP x — c z _ 3 ), and in general, 
(1) H^ n = «^_, (H x+ _ 1 + «P 2 — c z+tt _0. 

The above formula may be written : 

H^ n = u xJrn _ x (H^_, + tfP,)— ^ +B _ X X c x ^_ u 

r' d 

"We have previously represented — by ic xi and v-p by c z ; 

1 T C- I, 



therefore «,= , d*. Multiply both numerator and denominator 

by v ( noting that r'u is equal to 1), and we have 
r'v 1 L 



, xl x d x vl, — vd* 

v — rv y- V V-z- 



NOTES ON LIFE INSURANCE. 97 

Multiply the numerator and denominator of this fraction by r\ and 

i r% r'L r'L 

we have, u x = , ', = - — *— - — 1 
r vl—r'vd x l—d x 4+i 

r'l (J 

u x being found equal to —-, and c x equal tov-p, we have u x Xc x = 

4+1 4 

r'l m d x r'vdj x cl x „. . <L • -, • 

i — X v-jr = —, — j- = -j — • lhc expression - — = c x xu a is designat- 

4+1 4 4+l4 4+1 4+1 

ed by the symbol h x . 

Equation (1) then becomes, 

(2) II J+n = u x+n _, (ll x+n _ x + aV T ) — k x+n _ x . 

The expression for Jc x may assume various forms : 
/. _ d > _ 4— 4+i _h 1 _ h — -T,- 

*»+l <*+l **+l c x <■<■■» - 1 T" 



vr . r 



vroL — 1 = ; 1 — 7— — 1 = v- -. 1 — vu x — 1. 

1 v— 1 — r c x 1 — r c x 1 — r c x 

Referring again to i( x == j^- ; if both numerator and denominator 

4+i 

of this expression are multiplied by if* 1 , Ave have it x = - a * = 

v * 4+i 
v% D_ 



«»*+7 T) 

The numerical values of w, c, and h at each age have been calcu- 
lated and placed in columns, headed respectively u, c, and Jc. 

The quantities represented by the symbols u and k, although 

always retaining their correct numerical value, may come up in 

the general discussions in a variety of shapes. Sometimes these 

transformations are very convenient in the practical work of 

computing net values, as seen above in case of the factor u x = 

r' . D 
— becoming equal to =— ^- ; from which it is easy to obtain k, 

1 ?* C x -L'x+l 

by using the D column which has already been constructed. 

A THIRD GENERAL FORMULA FOR RESERVE. 

The deposit at the beginning of the nth. year being represented 
by H x+n _ lt the net annual premium by aP x , and the number living 
at the beginning of the year by 4+n-i- Assume that the number 
of insured persons is that given in the table. We will then have 



98 NOTES ON - LIFE INSURANCE. 

the net funds on hand at the beginning of the year represented by 
4+„-i Q&x+nr-i + aP x ). As before, call the ratio of interest r'. Then 
the amount that will be on hand at the end of the year will be ex- 
pressed by, 

Subtract from this the whole amount of death losses during the 
year, which is 4+ re -i — (4+«) ; because the amount insured in each 
case is $1, and the number of deaths during the year is equal to the 
number living at the beginning of the year minus the number living 
at the end of the year. The net fund remaining on hand at the end 
of the year, after the death losses are paid, is r'l x+n _ x (H a:+W _ 1 + 
aY x ) — (4+*_i — l x+n ). Divide this by the number living at the end 
of the year, and we will have the amount on deposit for each po- 
licy-holder at the end of the year. Therefore, 

it _ r'l x+n _, (H^^ + aP,) — (4+n-i — 4 +») .„ 

lx+n 
4+w [4+«— 1 4+w— 1 X T ( n x+n _i + U -T x ) j 

— 7 J 01 

''x+n 

= i _ksz!^i_ r ' (H, + ^ 1 +aP,)), or 

= l_!^i(l ! _(H^. 1+ aP.) 

r'l r'l 

But y~ ~ nas previously been shown equal to u x1 and — j 

4+i 4+» 

and since r'v = 1, and — = v, the equation becomes, 
(3) H x+n = 1 — u x+n _ x (v — aY x — Ha+nLi). 

Note. — It should be borne in mind that all the formulas and 
rules for determining the amount that should be in deposit at the 
end of any policy year are based npon the supposition that the net 
premiums the insurer has contracted to receive are those called for 
by the table of mortality and rate of interest designated. The 
amount insured has been assumed to be $1. 

The return premium plan insures the premium in addition to the 
$1. This somewhat complicated arrangement requires care in the 
application of the general formulas for calculating the amount that 
must be held in deposit. For instance, formula (2) previously de- 
duced shows that the net funds on hand at the beginning of any po- 
licy year, just after the net annual premium has been paid, must be 



+«— i 



NOTES ON LIFE INSURANCE. 1)9 

multiplied by u xi and h x subtracted from the product, in order to get 
the deposit at the end of that year. This k x is the amount that each 
policy-holder must contribute to pay losses by death during the year, 
on the supposition that the whole number of persons in the table 
were insured for $1 each. The return premium plan changes the 
amount insured from 81 to $1 plus the premiums paid. Therefore, 
after having multiplied the net funds on hand at the beginning of 
the year by u m9 it is necessary in this case to subtract k x times the 
amount actually insured, which, as stated before, is $1, plus the pre- 
miums paid. 



1 '. . 3JOTES OS IIFE Df3URA3fCE- 



CHAPTER YKL 

^•:~:t~5 ?ah ;rrz:"ij. tza>~ o>~:z a ti 



S tpposk that #1 Is to be paid quarter! y in advance for life. The 
value at age x of the four quarterly payments of $1 each, to be 

made the first year, is expressed b y P * +P ^^-^ P ^ . dthe 

supposition that the rate of mortality is uniform during the first 

1',-^1',-x D,-D,_ : -.=iD,-iI',_ : 

Therefore the above expression for the value at age as of the four 



■:r -^ — - — = — = == = — = — = — . or ^ — 



For the second year's quarterly payments of $1 each, the value at 
age x is expressed by 



- 1 :-.--':- .-!",- -.--'-- ' 



But we have, on the supposition that the rate of mortality is 
uniform during the second year . 

Therefore the foregoing expression for the value at age z of the 
four quarterly payments of $1 each, to be made during the second 
year, may be written : 

O- ----- -O- ---',- -3l'-.-fI'.-:-r-\-: 

o. g- , or g- ■ 



NOTES ON LIFE INSURANCE. 101 

In a similar manner we find that the value at age x of the four quar- 
terly payments of $1 each, to be made during the third year, is ex- 
pressed by *+2 4 *^ an £ go oi ^ j» ot ^^ success j ve y ear t0 t i ie 

table limit. Adding together all these respective yearly values, we 
have the numerator of this life series of quarterly payments of $1 
expressed by: J/D* + -^D x+1 + V D *+« + ¥ D *+3 + etc. By adding 
£D X to the numerator, it becomes ^N x , therefore the value of this 

series is expressed by: — — ^ — - — - = * x — J. If the quarterly 

payments are to be $J each, the foregoing expression is divided by 4 
in order to give the value at age x. Thus we have : 

vp — tV — fp — i- ^ tne ^ rst payment of $ J is deferred one 

N a N x 
term, the value becomes : =p — (|- + i) = fr f- 



In general, if the payment of $1 is made t times a year, the first 
payment in advance, the second at the end of a term expressed by 
one year divided by t, the third at the end of two of these terms, , 
and so on, we have for the value of these t payments of $1 each, for 
the first year : 

L\ + D a+ I_ + L\ + i.+ +D* + <_=J. 

It is assumed that the rate of mortality is uniform during any one 
year. The number of deaths will then be proportional to the time, 

and it follows that D. + JL = D."— \ (D x — D x+1 ) = * D *— P* + p *-m 

t t t 

finally D I + ^i = D,- f =i(D I -r> I+1 )= \t>, + ^D,*+i- The 

number of terms containing ~D X is equal to t. The first term is one 
time D x , which may be written -D x , the second term is — - D x , the 

t t 

f 2 1 

third term is D x , the last term is - D x . The sum of this arith- 

t t 

metical series - -I 1 -- + +- is obtained by the or- 

t t t t 

dinary rule for finding the sum of an arithmetical series of terms, 
namely, multiply the sum of the extremes by the number of terms, 



102 NOTES ON LIFE INSURANCE. 

and divide the product by 2. The sum of the extremes in this series 

is- H = ■ Multiply by t, we have ; dividing this by 2, 

t t t t 

we have = the number of times IX,. occurs in the expression 

that gives the value at age x of $1, paid t times the first year. But 

in the expression for the same year, we find L\ +1 enters a number 

X 2 t 1 

of times equal to t — 1. This series is - + — -f + The 

t t t 

sum of this series is ( 1 )x {t — l)-5-2 = Therefore, 

\t t I 2t 

2t represents that part of the value at age x of the t pay- 



ments of %\ each, made the first year, which is expressed in terms 

t l+l- D +^D. 
of D x+1 . Hence 2t 2t xl expresses the value at age x of 

K ' 

t payments of Si each, made the first year. 

For the second year, the quantity D* + i enters into the expression 
in a manner entirely similar to that in which D x enters into the ex- 

pression f or the first year. "We therefore find that 2t x ' 1 ex- 

presses the value at age x of that part of the t payments, made the 
second year, which is expressed in terms of D, +1 . That part of the 
first year's payments, expressed in terms of L\ + i, was previously 

f i 

found to be — — D i+1 . From which we have : 

Zt 

■ D x+1 + — -T> x+1 = — D z+1 = tD xM1 . 

2t + 2t ^ 2t + + 

This is the sum of all the terms of the general expression involving 
D x ^!, because none occur, except in the first and second years. In 
a maimer entirely similar, the sum of the terms involving D^ are 
found to be expressed by tD x+2i and so for D x+3 and all the Ds to the 
table limit. 

Referring to the terms involving D x , we find that these occur only 
in the first year, and that their sum has previously been found to be 

expressed by— — L\ = -- — ~D X ; therefore, the value at age x of this 

life series of payments of $1 each, made t times per year, the first 
in advance, is expressed by : 



NOTES ON LIFE INSURANCE. 103 



By adding — — D, to, the numerator, the first term becomes tD xi and 
the numerator will become £N X . Therefore, we have, 



the value at age a; of this life series of payments of $ 1 each made t 
times a year. Divide this by t, and we have, we have, 

3l — *— 

L\ ~.2* 

which is equal to the value at age x of a similar life series, but each 

payment equal to $— , amounting to $1 per year, but paid in £ install- 

ments at equal intervals, the first payment being made in advance. 

If the first payment of — is not made in advance, but is deferred one 

term, or a time equal to one year divided by t, the value of the life 

series is less by $- than the above expression will give. The expres- 

sion in this case is : 

ST, /*— 1 ,'1\ y, /*-! 2 _\_?i ___* + ! 
D s \M + t)~D x \2* + 2*/"~D a 2£ ° 



PART II 



PRACTICAL LIFE INSURANCE. 



" Consider for a moment the peculiar nature of Life Assurance. This is 
a business that presents the direct converse of ordinary commercial business. 
Ordinary commercial business, if legitimate, begins with a considerable invest- 
ment of capital, and the profits follow, perhaps, at a considerable distance. 
Bat here, on the contrary, you begin with receiving largely, and your liabili- 
ties are postponed to a distant date. Now, I dare say there are not many mem- 
bers of this House who know to what an extraordinary extent this is true, and, 
therefore, to what an extraordinary extent the public are dependent on the 
prudence, the high honor, and the character of those concerned in the manage- 
ment of these institutions." — Gladstone, 18G4. 

" Correct mortality tables and a safe rate of interest as a basis for rates of 
insurance, ample reserves to cover all contingencies, and sound and reliable as- 
sets, always available, out of which to pay obligations as they mature, are the 
corner-stones upon which life insurance rests. Lacking either, a company will 
sooner or later fail."— F. S. Winston, 1871. 



NOTES ON LIFE INSURANCE. 109 



CHAPTER IX. 

GENERAL MANAGEMENT. 

In order to pay the expenses of conducting the business, it is 
necessary that additional means should be provided over and above 
the net premiums ; the latter being enough, and only enough, on 
the data designated by the State, to pay the cost of insurance, and 
furnish the requisite deposit or " reserve.'" 

It is usual to add to the net premium from twenty to thirty per 
cent, or even more, for the purpose of defraying expenses. This 
addition to the net premium is technically sailed loading. 

The loading may, and often does, more than pay expenses. 

The interest actually received is nearly always more than the net 
interest assumed in the table calculations. 

And the actual mortality, particularly in the earlier years of a 
company, is, in practice, generally less than that given in the^tsMe. 

From each of the three above-named sources, surplus may be ob- 
tained. By surplus is here meant money, or its equivalent, in ex- 
cess of what is required to pay losses by death during the year, to 
form the " deposit " for the policy at the end of the year, and pay 
all expenses. The surplus, in purely mutual companies, belongs to 
the policy-holders. In the purely stock companies, all the surplus 
goes to the share-holders. The mixed companies are those stock 
companies that give some portion of the surplus to the policy- 
holders. 

In order to investigate the nature of practical Life Insurance 
business for one year, let us suppose that the cost of insurance, and 
all expenses of the previous year, have been paid, and that the com- 
pany had on hand, at the close of the previous year, the requisite 
deposit for each and all of its outstanding policies. We will, for 
the present, suppose that the surplus of the previous year had been 
distributed to its respective owners. 

At the beginning of the year, the business of which we are now 
investigating, each policy-holder pays his full annual premium. 
There is then in the hands of the company, on account and to the 
credit of each policy, the two amounts — namely, the deposit at the 
end of the preceding year, and the full annual premium. 



HO NOTES ON LIFE INSURANCE. 

These sums are both invested at the best, safe rate of interest ; 
and out of these two amounts, thus increased by interest, actually 
received during the year, the " expenses " for the year, properly 
chargeable to each policy, must be paid ; the cost of i lsurance, or 
proportion of losses by death during the year, properly chargeable 
to each policy, must be paid ; and the requisite deposit at the end 
of the year for each policy must be securely invested for the policy- 
holder at the net or table rate of interest at least. If there is any 
thing left on account of each policy, it is surplus produced by the 
policy. 

When the surplus arising from the funds of each policy is ob- 
tained as above, and is distributed in accordance with this principle, 
it is said to be divided upon the " contribution plan." 

Let us now consider the loading which has been added to the 
net annual premium, and the expense which this loading is intend- 
ed to provide for. In the first place, it may be remarked that the 
" expenses" properly chargeable to a policy, are not necessarily the 
same proportion of the annual premium in different cases. At the 
end of the year, although it may require some labor to adjust with 
precision the expense account for each separate policy, or each dis- 
tinctive set of policies, this should be attempted, and substantial 
equity in this respect can always be attained. The amount charged 
any year to a policy on account of general expenses should be in 
proportion to the amount of insurance the company furnishes that 
year to the holder of the policy — that is, the amount called for by 
the policy less the deposit. 

In favorable or even in ordinary years, the loading and the 
interest on the funds of the company (because of their realizing 
usually a higher rate than that called for by the table calculations) 
will produce a " surplus " on each policy at the end of the business 
of a year. This surplus arises from previous over-payment in ad- 
vance, demanded by the company, in order to make the business 
safe in the worst year that may occur in a lifetime. The surplus 
distributed to policy-holders is merely a return to them of that part 
of the premium they paid at the beginning of the year, which, at 
the end of the year, is found not to have been required during the 
year, either in effecting the insurance, providing the means (the de- 
posit) for paying the policy at maturity, or in paying " expenses." 

In Life Insurance, there are peculiar and mandatory arithmetical 
laws by which particular money values are computed — in addition, 
and after these values are accurately determined — practical Life In- 
surance becomes like all other business which involves the handling 



NOTES ON LIFE INSURANCE. Ill 

and control of vast amounts of money. Good judgment, great 
industry, the strictest integrity, and sound practical business sense 
are all absolutely essential to successful management. 

No prudent man will ever attempt to control or conduct any im- 
portant business without making some kind of estimate in advance. 
The mortality table furnishes the means for making certain esti- 
mates with an accuracy that is not usually found in ordinary busi- 
ness. But the " expenses " that will be incurred, or the rate of 
interest that will be realized on the investments, or the bad invest- 
ments that may be made, or whether some of its officers may not 
prove to be dishonest, and a variety of highly important questions 
of this nature, can not be settled by estimates made beforehand by 
life insurance companies, any more definitely than similar estimates 
can be made in any other business. Nevertheless, these estimates 
of practical results ought always to be made in advance. 

Some companies assume at the beginning of a year, that the busi- 
ness during the year will be such that they can safely deduct a cer- 
tain per cent from the premium. This deduction is miscalled a 
dividend. Business men understand that dividends are paid only 
out of earned net profits. They would be shocked at the idea of 
dividends paid on an assumption in regard to future profits, and 
consider it a fraud for the shareholders of a company to declare 
a dividend upon the stock when the capital is impaired. 

The subject of accounts in life insurance companies will never 
be definitely settled, until the book-keepers and accountants clearly 
understand the theory and principles upon which life insurance is 
founded. It is safe to say, that if any money account is kept with 
a policy at all, it ought to be correct. 

Illustrative Example. — The following arithmetical example is given 
in illustration of the accounts of a policy for any year : 

It is assumed that an ordinary whole-life policy for one thousand 
dollars, taken out at age forty-two, is in its tenth year. The net 
annual premium (Actuaries' 4 per cent), as previously calculated, is 
$25.55 ; take the loading to be 33 J per cent of this, then the fall 
annual premium is $34.05. To make out the account of this policy 
during the tenth year, we will assume that the expenses properly 
chargeable to it during the year are, twenty per cent of the full or 
gross annual premium ; that every thing to the credit of this policy 
at the end of the preceding year, except the deposit, had been dis- 
tributed to its owner : that the rate of interest actually realized by 
the company on its aggregate investments during the year was 



112 NOTES ON LIFE INSURANCE. 

seven per cent ; and that the mortality amongst the insured during 
the year was that called for by the table. 

The deposit for this policy at the end of the ninth or the begin- 
ning of the tenth year is $156.33. From the gross premium, $34.05 
paid at the beginning of the tenth year, deduct twenty per cent for 
expenses, and we have left $27.24 ; add this to the deposit, and we 
have $183.57 at the beginning of the year to the credit of the 
policy, after having provided for expenses. 

This increased by seven per cent during the year amounts at the 
end of the year to $196.42. The amount that must be on deposit at 
the end of the year is $175.16. The cost of insurance on the amount 
at risk during the year is $13.93. After this is paid, and the deposit 
at the end of the year is set apart, there will be $7.33 surplus on 
hand. This is about twenty-one per cent of the gross annual pre- 
mium paid at the beginning of the year. If the company returns it 
to the policy-holder, this may prove that he paid at the beginning 
of the year more than was necessary ; but in a business sense, it can 
not be maintained that the policy-holder invested the $34.05 at 
twenty-one per cent per annum. 

In case the expenses for the year and the mortality amongst the 
insured had been greater than that assumed in this example, and 
the interest had been less, this surplus would have been diminished. 
On the other hand, had the variation in expenses, mortality, and 
interest been the opposite of the above, the surplus would have 
been greater. We have seen, that with a loading of 3 3|- per cent 
on the net annual premium, there was, at the end of the year, a sur- 
plus of $7.33 : no great margin, token the question is that of the 
prompt and certain payment at maturity of a policy of one thousand 
dollars, more especially in case the surplus or over-payment made at 
the beginning of the year, in order to make the payment of the policy 
safe, is returned to the policy-holder at the end of the year. 

When the surplus belonging to the policy-holder is not distributed, 
but remains in the hands of the company to the credit of the policy 
that produced it, it ought to be invested for the holder of the policy. 
"When the surplus has all been distributed, the true value of the 
policy at the end of any year, and before the payment of the next 
annual premium, is the deposit ; but when it has not been distributed, 
the true value of the policy is the deposit, plus any surplus there may 
be in the hands of the company to the credit of the policy. 

When the surplus is distributed to the policy-holders, it may be 
used in part payment of the next annual premium, or it may he 
applied to the purchase of additional full-paid insurance. The latter 



NOTES ON LIFE INSURANCE. 113 

would progressively increase the amount of the policy ; the former 
would diminish the annual premium. 

Reversionary Value. — When the amount of surplus to be returned 
lias been determined, the amount of full-paid insurance that this 
surplus will purchase at that age is calculated by first finding the 
net single premium that will insure one dollar at that age. 

For instance, suppose that at age 30 the surplus is 115.36 (be- 
sides a loading for expenses). The net single premium that will at 
that age insure one dollar for whole life is $0.300158 ; and the ques- 
tion simply is, if $0.306158 will insure one dollar for whole life, how 
much will $15.36 insure ? 

By solving this simple proportion, we find the amount is $50.17; 
and this is the addition to the policy that the surplus named will 
purchase. This additional insurance is full paid, and the $50.17, in 
this case, is called by insurance writers the reversionary value of the 
surplus, $15.36. 

Any proceeds that may in the future arise from interest on this 
$15.36, in excess of the four per cent necessary to pay the cost of 
insurance, pay expenses, and provide the requisite deposit, will be 
additional surplus, and may be used as it accrues in purchasing ad- 
ditional full-paid insurance. 

Premium Notes. — A life insurance company can, with safety to 
itself, accept the notes of a policy-holder in part payment of the 
" net annual premium," and the amount of these " notes " or " loans" 
bearing net or table interest, may equal, but must not exceed, the 
deposit. The deposit increases from year to year, and the notes or 
loans may be increased to the same extent, but no more. The notes 
or loans must be deducted from the face of the policy at maturity ; 
therefore, the amount actually insured becomes less and less each 
year. The question is not, " Can a life insurance company safely 
accept notes in part payment of the annual premiums ?" but rather, 
" Can a policy-holder, for any great length of time, afford to accept 
the credit proffered by the company ?" 

Suppose that we take this case to the limit of the table, ninety- 
nine years. The policy-holder will have paid, each year, his propor- 
tion of the losses by death, and the yearly expenses ; and the de- 
posit, consisting entirely of his own notes, will have amounted to 
within a very small fraction of the whole amount of the face of his 
policy. The man dies in the one hundredth year of his age, and the 
heirs receive his notes in part payment of the policy ; and these notes 



114 NOTES ON LIFE INSURANCE. 

are, in this particular case, enough to fully pay the policy when the 
last annual payment only, in money, is added to these notes. 

This certainly is not a desirable kind of life insurance for those 
who live long. On the other hand, if the insured dies early, he will 
gain by the note or loan system. 

The life insurance company is safe in this case, provided it has 
a large number of policy-holders, and retains them to the end of 
their lives. 

It is true that note or loan companies seldom, if ever, in 
practice, push the credit system to the extreme limit given above ; 
but they may do it with safety under the above proviso. The ques- 
tion is, can the policy-holder stand it if he does not die soon ? 

The complication in the accounts of a company arising from the 
note system, when carried into a large number of policies, results in 
great confusion and irregularities. The better opinion seems to be ? 
that it will be to the ultimate advantage of companies and policy- 
holders if this system of credits in life insurance by notes, loans, 
or other devices is abandoned, or, at least, brought within very nar- 
row limits. 

This note or loan system of life insurance has strong advocates 
amongst well-informed insurance writers. But in the long run, po- 
licv-holders will find there is some delusion about the credit so gen- 
erously proffered and urged upon their acceptance. It is true that 
if a man is certain that he will die soon, and he can get $100 worth 
of insurance for $50 in cash and his note for $50, he would do well 
to take out a policy in a note company, die during the year, and let 
his heirs receive the amount of the policy, less his note for $50 ; but 
there are strong reasons why the system of note or loan life in- 
surance is not advantageous to those who continue to renew their 
policies in such companies for any great length of time. 

Stock and Mutual Mates.!— Purely stock companies are those in 
which all the surplus belongs to the shareholders. Such companies 
seldom, if ever, accept notes or give credit in part payment of pre- 
miums. As a general rule, they charge less than the purely mutual 
or mixed companies. Their theory is, that they make divide /ids to 
policy-holders in advance by charging less premiums. The fact is. 
that dividends to holders of life insurance policies are simply a re- 
turn of that part of the annual premium which was paid to the 
company at the beginning of the year, and which, at the end of the 
business of the vear, is found not to have been required in paying 



NOTES ON LIFE INSURANCE. 115 

the expenses, paying the losses by death, and providing the requisite 
deposit at the end of the year. 

Mutual companies often abate, say twenty or thirty per cent, 
more or less, from the annual premium called for by the policy. 
This practically reduces the premium for the year, but can not fairly 
be called a dividend in the sense of income from premiums invested 
in life insurance. In some cases, these deductions have been nearly 
uniform for a period of years. 

Stability of Companies. — Notwithstanding the correctness of the 
theory upon which the business of life insurance is founded, assum- 
ing that the table of mortality is accurate, and that net interest is 
always realized, there are many contingencies that may prove fatal 
to companies in practice ; and whilst strict compliance with certain 
fixed principles and definite rules will always enable a company to 
pay its policies at maturity, there are many things that will, if per- 
mitted to occur, bankrupt a life insurance company. These compa- 
nies are not exempt from the effects produced by dishonesty, fraud, 
and defalcation. Moreover, continued lavish expenditures, the selec- 
tion of bad risks by insuring impaired or unhealthy lives, or making 
unsafe investments, will result in disaster. 

There can scarcely be any saying more groundless than the state- 
ment often heard, that " life insurance companies can not break." 
And, on the other hand, it is absurd to say, that, when well con- 
ducted in every particular, it is impossible for life insurance com- 
panies to comply with all their obligations, and pay all their policies 
at maturity. The plain fact is, that life insurance companies can 
break, and tcill break, unless managed with skill and integrity. On 
the other hand, it is undoubtedly true that the business of life insur- 
ance can be made more secure than any other commercial business 
known amongst men ; and whilst it may be made the safest, it is 
a business in which, if it is not thoroughly comprehended and 
strictly guarded, designing fraud may raise a curtain, behind which 
the worst schemes can be carried on free from detection, until such 
time as the death claims exceed the annual premiums ; that is to 
say, for thirty or forty years. 

To f ulry appreciate this fact, it is only necessary to recall the illus- 
tration previously given, in which it was seen, that, at the end of 
the thirty-fourth year, nearly $28,000,000 was on hand in deposit, 
after paying all the death claims that had previously matured. This 
sum, and all the future net annual premiums, with compound inte- 
rest on the whole, is required in order to enable the company to meet 



116 NOTES ON LIFE INSURANCE. 

its liabilities. Suppose that this -$28,000,000 had been appropriated 
to other purposes ? This might have been done, and the company 
have paid all its losses up to that time, paid all expenses out of the 
loading, and, to external appearance, have seemed all right ; and 
this, too, with a real defalcation of $28,000,000. 

It is essential to the policy-holder that the life insurance com- 
pany with which he may take out a policy, should be controlled 
by wise and stringent laws, rigidly enforced ; because, from the na- 
ture of this business, the funds held in trust are peculiarly liable to 
misapplication. To insure safety in the business, every detail 
should be furnished, at least once in every year, to some competent 
State officer ; and by the latter the accounts should all be carefully 
recomputed, and the results published. Sound and well-conducted 
companies desire this, and others should be forced to a full exhibit 
of all their affairs. 

Conditions contained in the Policy. — If the officers are, in every 
respect, the right men for this most important and gigantic business, 
it is well to look further and inquire closely into the terms and 
conditions of the contract between the company and the policy- 
holder. These are expressed in the policy, and in some companies 
are liberal and just ; in others they are harshly restrictive, not to 
say unjust. It is but a few years since it was the universal practice 
of life insurance companies to appropriate to themselves the whole 
accrued value of a policy in case the holder thereof failed on a given 
day to pay his annual premium. 

There was no justification or excuse for this rule of forfeiture for 
non-payment of premiums except that this was a condition expressed 
in the contract. That it continued for so long a time to be the uni- 
versal custom can only be accounted for by the fact that the princi- 
ples upon which life insurance is founded were not thoroughly un- 
derstood by business men. There can be no safety or certainty of 
the payment of policies at maturity, and, therefore, no real insur- 
ance, in case a company charges less than than the net annual pre- 
mium, and a loading sufficient to cover expenses. Because, in spite 
of all we hear about large " dividends''' to policy-holders, arising 
from the " investment" the net annual premium and net* interest 
upon it must go to effect the insurance, and the expenses must be 
paid in addition. It appears, from this view of the case, that a life 
insurance company may charge too little. 



NOTES ON LIFE INSURANCE. 117 

Certainty of the Payment of his Policy at Maturity is what every 
Policy-holder icants. — To insure this, it is necessary that the compa- 
ny should charge enough to enable it to meet all its liabilities dur- 
ing the worst year that may reasonably be expected to occur during 
the continuance of the contract ; and this is generally for a life- 
time. Therefore, when the mortality is greatest, and the interest 
on investments lowest, and expenses heaviest, the company must 
have the means of meeting its liabilities. It follows that, in favo- 
rable years, there will be an over-payment. In case this over-pay- 
ment is all returned to the policy-holder at the end of the business 
of the year, it is not a matter of vital importance whether the pre- 
mium is a little more or a little less, provided it is enough to make 
the payment of the policy at maturity certain. 

Numerical Bragging. — The expenses of life insurance companies 
are large. Agents' commissions, salaries of officers, traveling ex- 
penses, taxes, printing, rents, stationery — these and other expenses 
have to be paid in cash. The losses that occur during the year, by 
death, must be paid. The expenses and the losses by death are 
paid by the company ; but this is clone with the money of the policy- 
holders. 

The deposit is a specific amount, determined by accurate arith- 
metical calculation ; this amount must be in the hands of the com- 
pany, and held securely invested at a certain rate of interest, and 
this interest regularly compounded every year, in order to enable 
the company to pay its policies at maturity. The company must 
retain the deposit for each and every one of its outstanding policies ; 
must pay current expenses; must pay the losses that occur by 
death, each year, of a certain number of policy-holders ; and as the 
company can only make seven or eight per cent by safe investments 
of the funds intrusted to it by the policy-holders, the u enormous 
dividends 1 ' 1 so much talked of may well be styled " numerical brag- 
ging: 1 

Method of Calculating Net Values should be understood. — The 
mere fact that a man can compute interest on money will not make 
him a competent banker, neither will a knowledge of the formulas 
and rules be in itself sufficient to fit one for the important business 
of life insurance. But it would be far better to intrust banking to 
men who can not calculate interest on money, than to intrust life 
insurance to those who are not acquainted with the method upon 



118 NOTES OX LIFE INSURANCE. 

"which calculations of important money-values in this business are 
based. 

There is danger to all in the doctrine, often promulgated by com- 
panies and agents, that life insurance business can be better con- 
ducted by men who do not understand the " method of calculat- 
ing these values" than by those who do understand the simple 
principles upon which alone this business can be safely conducted. 
Those who talk in this way are, generally speaking, "forty per cent 
dividend men" who propose to lend one third or one half the pre- 
mium to the policy-holder at six per cent, and promise him forty 
per cent dividend per annum upon the whole amount of the premi- 
um. The same persons generally style the money of the policy- 
holder that is held by the company in trust for the purpose of ena- 
bling it to pay the policy at maturity, " cash capital" or, at least, 
announce millions of assets, and are silent about these assets being a 
deposit debt, held by the company in trust for policy-holders. 

Medical Examiners. — The general law governing the duration of 
human life will be of little or no avail in case a life insurance com- 
pany accepts risks upon impaired or diseased lives ; and companies 
that have only a small number of policy-holders will always be, to 
some extent, liable to a number of losses not in accordance with the 
general law of duration of human life ; because this law only applies 
to a large number of selected lives, not to a single individual, or to 
a small number of individuals. Much of the success of life insur- 
ance companies depends upon the skill and integrity of the medical 
examiners. 

It is worthy of notice, too, that if the number of deaths in any 
one year should prove to be remarkably small, it is not safe to as- 
sume that, because the losses by death in that year are greatly less 
than those called for by the Table of Mortality, the difference is 
clear gain, and can be disposed of as " surplws" and distributed at 
the end of the year ; because the variation from the number of 
deaths called for by the Table of Mortality will probably soon vary 
on the other side. These losses have to be paid, and that promptly. 

Besides variations from the table rate of mortality that may and 
do occur in practice, it should be noticed that in case a policy for 
$100,000 is grouped with ninety-nine others of $1000 each, the death 
of this single individual would be a greater loss to the company 
than that of the other ninety-nine policy-holders. These things and 
many others of a similar nature have to be closely watched. 



NOTES ON LIFE INSUKANCE. 119 

Comparison between the Mortality experienced and that called for 
by the Table. — To compare, at the end of any calendar year, the 
mortality actually experienced by a company during that year, with 
that called for by the table of mortality used in computing net 
premiums, the following method may be and often is used. The 
results obtained, though not theoretically exact, seldom in prac- 
tice involve appreciable error. Take all the insured in the com- 
pany that attained, during the year, any named age. These are 
assumed to be exposed to the same risk. Those that were insured 
only for portions of the year are treated as so many fractions of a 
year. In illustration, take age 40; suppose that the number who 
reached this age during the year just passed is 120; and that of this 
number 100 have been insured during the Avhole year, five for three 
fourths of the year each, ten for half the year, three for one fourth 
of the year, and two for one twelfth of the year. We have 100 + 
"V+'V^ + f + T 2 ^=109|= number of lives at risk for the whole year 
that reached forty years of age. The American Experience Table 
of Mortality shows that of 7S,862 living at age 39, 756 of 
these will die before they reach age 40. Therefore the number 
of deaths to be expected in this case, as shown by the table of mor- 

. -, , 756X1094 
tality we are now using, is expressed by — — - = 1.051305. If 

78862 

the actual mortality is less than the above, the result for this age is 
favorable ; and if more it is unfavorable. 

By a similar process, compare the deaths at each age with those 
called for by the mortality table, and the general result will show 
whether the actual number of deaths amongst all the insured in the 
company during the year just passed is greater or less than the 
number indicated by the table used. 

It is not enough to know how the actual death rate compares 
with that of the table ; because those who die in the year may be 
insured for more or less than the average policy in the company. 
To make an estimate of the whole amount of claims that should 
have accrued during the year for death losses on the original as- 
sumptions, it will be necessary to ascertain at each age the average 
amount at that age on which a year's risk of mortality has occurred, 
and multiply this amount by the table rate of mortality at the age. 

The comparison of actual results with table assumptions, in regard 
both to number of deaths and amount of death claims, should be 
made in each company at least once a year. To facilitate these 
computations, tables have been constructed, that give in decimals the 
fraction of a year, from any day on which a policy may be issued 



120 NOTES OX LIFE INSURANCE. 

to the end of the calendar year, and others giving the percentage 
of deaths to number living at each age in the mortality table. 

Life Companies great Money Lenders. — It is often urged, that life 
insurance companies are absorbing a very large portion of the cur- 
rency of the country ; and many persons seem to apprehend that 
this will result in extraordinary scarcity of money. But it must be 
remembered that life insurance companies are compelled to keep 
their funds constantly invested ; they are, therefore, forced to be 
lenders of money ; and, as a general rule, they are more careful 
about the character of their securities than anxious to realize exorbi- 
tant rates of interest. Some of the States have passed laws requir- 
ing their companies to invest exclusively in securities in their own 
State or in United States bonds. These restrictive laws are unjust 
to the citizens of other States, policy-holders of these companies. 
A company should be allowed — if not peremptorily required — to in- 
vest in any State the net funds received from citizens of that State. 

Campaign Literature. — The large per cent of the premiums paid 
to agents is an item of very heavy expense to life companies ; and 
another great expense is the publishing of a large amount of what 
is called i: campaign literature." It is perhaps impracticable for the 
companies to materially lessen these enormous expenses, so long as 
the present extraordinary competition is kept up, and the public arc 
not informed in regard to the true principles upon t business 

ought to be conduct*: J. 

If policy-holders had clear and distinct ideas of their own in re- 
gard to life insurance, and would seek for the best article at a fair 
price, as they already do in regard to their other purchases, the 
companies would no doubt be but too glad to abate from their pre- 
miums that portion of the " loading " which now goes to pay these 
large commissions to agents and publish i; campaign literature.*' 

He-insuring. — Existing laws in most of the States authorize in- 
surance companies to re-insure any of their risks, or any part there- 
of. This was no doubt intended to apply to cases in which a 
pany might have an opportunity to insure more upon one risk than 
it would, in the opinion of its officers, be justified in carrying. 
For instance, a person might desire to insure his life for $20,000. 
This risk, say, is taken by a company that can safely carry only 
$5000 on any one life. The company is by law authorized to place 
in other companies the remaining $15,000. or even the whole $20,- 
000. This is done without consulting the policy-holder, or intimat- 



NOTES ON LIFE INSURANCE. 121 

ing to him that the company unaided does not feel safe in carrying 
the $20,000. The law in this respect, whilst very convenient for 
companies and advantageous to the agents who receive commissions 
on these large amounts, is hardly fair to policy-holders. 

But when permission to reinsure any risk is used for the purpose 
of wholesale transfer of * all the policy-holders of one company to 
another, without the knowledge or consent of the insured, and the 
interests of managers are alone consulted, reinsuring becomes in 
many cases a great evil, resulting in wholesale amalgamations, disas- 
ter, and ruin on a large scale. 

To prevent this, it has been strongly urged that life insurance 
companies be prohibited from reinsuring any of their risks, or any 
part thereof, without the written consent of the policy-holder. 
Against this prohibition great outcry is made by many on the as- 
sumed ground that the interests of the policy-holders would often 
be sacrificed because companies are not allowed to transfer them by 
wholesale to the highest bidder. It has been well said that " the 
evil resulting from the power to reinsure given to life companies 
more than counterbalances the good possibly inherent in the exercise 
of that power." In admitting this, some over-zealous friends of 
policy-holders still insist that companies should be allowed to rein- 
sure without consulting the insured. It required the consent of both' 
parties to make the contract, and it is but fair that the consent of 
the insured should be obtained before he is traded off by the com- 
pany that contracted with him. 

On no account should any life insurance company be permitted to 
transfer its policy-holders or any portion of them to another com- 
pany, and this company allowed to issue its own policies in lieu of 
those formerly issued by the first company, the new policy bearing 
the date of the transfer. This should not be permitted even with 
the written consent of the policy-holder, because it is a fraud for 
the purpose of escaping, on the part of the second company, liabili- 
ty for the accrued net value of the original policy. 

Suppose, for instance, that a company has been doing business for 
ten years, it has 10,000 policies, the aggregate deposit or accrued 
net value of these policies being, say, $3,000,000. If all these poli- 
cies are taken up by the second company, and new policies issued 
containing the same terms, except the date is that of the transfer in- 
stead of being that at which the policy was originally issued, the 
accrued liability is lost out of the accounts, and the manipulators of 
this transaction can at once appropriate the deposit of $3,000,000, 
leaving the future to take care of itself. 



122 NOTES ON LIFE INSURANCE. 



CHAPTER X. 

VARIETY IN PLANS OF INSURANCE. 

It has been recently stated on good authority, that " some com- 
panies in their prospectuses propose to issue as many as eight or 
nine hundred varieties of policies, each of which would require a 
distinct table of surrender values." This must be understood to 
apply to the length of time for which insurance is effected, as well 
as to differences of general plan. 

The Superintendent of the Insurance Department of New-York, 
in his report, 1870, says : " It is believed to be a fact now causing 
quite general complaint, that there are too many complicated 
schemes or plans of insuring, as well as too many and too elaborate 
forms of contract or policy. It is difficult to perceive any excuse 
for the promulgation of so many theories and schemes, except upon 
the ground that they are intended to accomplish just what is ac- 
complished, to wit, the entering into contracts by the insured, the 
true force and effect of which they do not understand." 

It is suggested that life insurance companies and the actuaries 
should use their influence to lessen instead of increasing the num- 
ber of plans and schemes, and endeavor to impart to the educated 
public correct and practical knowledge of the simple principles 
upon which true life insurance is founded. 

Insurance for one year only. — The first question is the price to 
be paid at the beginning of the year for each $1 of insurance pur- 
chased by the policy-holder. 

It is usual to assume that a person who applies for insurance, is 
exactly a given number of years old. The mortality tables and the 
calculations are based upon whole years ; and the age is taken to 
be the whole number of years nearest to the real age. For 
instance, if the real age of a person is thirty years and five months, 
he is considered thirty years old ; but if the real age is thirty years 
and seven months, he is taken as thirty-one years old. 

Although in theory the amount of a policy is not due until the 
end of the policy year within which the insured may die, it is usual 
for life insurance companies, in practice, to pay the j>olicy within 



NOTES ON LIFE INSURANCE. 123 

from thirty to ninety days after proof of the death of the 
insured. 

In case of insurance for one year only, the net amount that must 
be paid at the beginning of the year to insure $1 to the heirs of the 
insured at the end of the year, provided he dies before that time, 
is calculated in the following manner. 

Notice that if the insured does not die, the $1 is not to be paid to 
him or his heirs, and that the premium paid for this insurance is 
gone ; not lost, however, but paid out by the insured for insurance 
on his life for one year. 

To obtain at any age the amount that will insure $1000, to be 
paid to the heirs of the insured at the end of one year, in case the 
insured dies during the year : a table showing the rate of mortality 
must be furnished, and a rate of interest fixed upon. Assume that 
the table is that which purports to give the rate of mortality 
among insured lives in this country, which is called, American Ex- 
perience Table of Mortality. (See page 15.) Suppose the interest 
is assumed to be seven per cent, and that the person to be insur- 
ed for one year is aged 50. The amount that will, if paid in ad- 
vance, and invested at seven per cent, produce $1 certain in one 
year, w T hen principal and interest at this rate for one year are 
added together, is obtained by dividing 100 by 107. This makes 
$0.934579. Then multiply this amount by the number of deaths 
given in the table opposite to age 50, which is 962, and divide 
the product by the number living at the same age, which is 69,804. 
The result is $0.012879. This is the amount that will insure $1 for 
one year, if paid in hand at age 50. One thousand times this 
amount, or $12.88, will insure $1000 for one year at the same age. 
In a precisely similar manner, the calculations are made for in- 
surance for one year at any age, and for any amount, and at any 
rate of interest. 



Note. The rate of interest that may be realized for one year must be judged of by the par- 
ties to the contract— namely, the insurer and the insured. 

The mortality table is actuarial work— that is to say, the actuaries collect and arrange the 
statistics, and from observation of the death-rate deduce a table for practical use. In reference 
to tables of mortality, the distinguished Professor Edward Sang, of Edinburgh, says (in 1864), 
"The smoothing, as it is called, of a life-table is always to be deprecated; we can only judge of 
the propriety of smoothing by comparison with some table which we deem more trustworthy, 
but we ought to adopt that which is more deserving of confidence." 

The differences in the tables now mostly used in this country are not so great as to be of 
much consequence in practice. None of them are supposed to express with perfect accuracy 
the law of duration of human life. Even if they did so express this law, there would be no 
certainty in advance that this law of duration would always apply to the insured lives in each 
company. 



124 NOTES ON LIFE INSURANCE. 

If the insurer can make only six per cent on the money during 
the year, the net amount that would have to be paid for the in- 
surance in this case is greater than that in the foregoing example, 
in which the rate of interest is assumed to be seven per cent ; be- 
cause the amount of money necessary, at six per cent, to produce 
$1 in one year, is greater than the amount that will at seven per 
cent produce $1 in the same length of time. Whatever may be the 
rate of interest assumed, the insured can readily calculate as above 
the net price of his insurance for one year on the designated table 
of mortality, and at any named rate of interest. In this net price 
no allowance has been made for expenses or profits. Business men 
usually know something about expenses in general ; and after get- 
ting at the above net price y they may form some idea of what the 
expense of this transaction ought to be to the company. In addi- 
tion to expenses, there must be some margin for profit to the in- 
surer, otherwise capital would not engage in the business. 

From the above calculation, interest being seven per cent, we 
find that at age 50 it takes $128.79 net price to insure $10,000 for 
one year. Add to this say fifteen per cent for expenses and ten per 
cent for profits, and we have $160.99 full premium. 

By comparing this with the premium charged by a company, an 
idea can be formed of the margin for contingencies and profits. On 
this plan, the policy-holder pays for insurance for one year only ; if 
he does not die during the year, his premium goes to pay the poli- 
cies of those that did die, and he has nothing. The objection to 
this plan is, that these yearly payments gradually increase until at 
the table limit, the net price of insurance for one year is the amount 
that will at net or table interest produce in one year the amount 
insured. 

Another objection is, that the insured may not be able to pass 
the medical examination at the beginning of each following year. 
Medical examinations every year are expensive as well as vexa- 
tious. This kind of insurance at the younger ages is cheap, but 
not generally desirable for the reasons above given. Still there are 
many cases in which insurance for a short term may be advanta- 
geous. 

Insurance for Whole Life paid for in Advance. — The net price 
in this case, as previously explained, is obtained from the commuta- 
tion-tables by dividing M, at the age of the applicant for insur- 
ance, by D at the same age. The net single premium that will at 
age 50 insure $10,000 for whole life is $4300.37 



NOTES ON" LIFE INSURANCE. 125 

We have just previously seen that at age 50 the net price, inte- 
rest being seven per cent, for insuring $10,000 for one year is 8128.79. 
It is supposed that a man insures his life because lie desires to leave 
money to his heirs in case of his death. There is no certainty 
that any individual will live for any named length of time, no mat- 
ter how short that time may be. Suppose he insures his life for 
$10,000 for one year only, at age 50, as above, and dies during the 
year; his heirs get the $10,000. The net price is $128.79. But 
suppose he insures upon the net single premium plan for whole life 
and dies during the first year ; his heirs would get the $10,000, 
but the net price is $4300.37. The net cost to the insurer is the 
same in each case, but the insured has paid $4300.37 for an amount 
that he might have secured to his heirs by the payment of $128.79. 
If he had not died before the end of the first year, the $12S.79 he 
paid would have been gone, and he would not have been insured 
after the first year; whereas, the payment of $4300.37 effected his 
insurance for whole life. It is not easy to see why a person should 
desire to pay for whole-life insurance by a single premium in ad- 
vance, if an arrangement can be made by which he can be certain that 
the insurance will continue for whole life, paid for by installments. 

Insurance for Whole Lifepaidfor by net Annual Premiums. — By 
reference to the table (page 32), it will be seen that, at age 50 (Am. 
Ex. 4£), the net annual premium that will insure $1000 for whole 
life is $32,490. Therefore, $324.90 is the net annual premium for a 
similar policy for $10,000. The net premium at age 50 for insuring 
$10,000 for one year only is $131.88 (Am. Ex. 4-J-. See page 19). 
Therefore, in case the insured pays $324.90 net annual premium at 
the beginning of the first year, he pays for more insurance than he 
gets from the company during that year ; this overpayment is 
placed to his credit and forms the deposit at the end of the year 
for his policy. This plan is a medium between insurance for one 
year only and that for whole life by a net single premium. 

Insurance for a long term of years, or for whole life, paid for in a 
limited number of years by equal annual premiums, partakes in a 
modified degree of the plan by which the whole insurance is effected 
by a single premium in advance. This plan may be advantageous 
in case the insured wants to pay fast and largely, in order to get 
through sooner than he would by paying less each year, but continu- 
ing to pay for a greater number of years. 

The Decreasing Annual Premium Plan may be advantageous in 
case the insured desires to pay excessively the first year in order 



126 



NOTES ON LIFE IXSUEAXCE. 



that his payment may be le ; - the second year, but still pay exces- 
sively the second year in order that his payment for the third year 
may be less than it was the second: and so on, decreasing each year. 

Tl\c Return Premium Plan. — A glance at the following table 
will show what these premiums must be at the different ages in or- 
der to carry out a contract to insure |1000 for whole life, and re- 
turn all the premiums at death, without interest, and will show at 
the different ages the amount of insurance for each age that might 
be purchased with the same money under a contract with different 
conditions : 




Insu xfor a Term of Years ar.'Z \ -ment at the end of the 
Term. — Strictly speaking, the ordinary life policies are, in fact, en- 
dowments at age 100 by the Actuaries' table, and at 96 by the 
American Experience. In the former, it is assumed that all living 
at age 99 will die before they reach 100, and in the latter that all 
living at age 95 will die before they are 96. It has been recom- 
mended with good reason that insurance upon lives in general 
should not extend beyond age 75, and suggested that endowment 
at that age be combined with insurance to that time. This su__ - 
tion. if adopted, would make the policy more costly, but would have 
the advantage of conferring upon the person who bad paid the pre- 
miums means for his own use hi case he survived the time at which 
men generally are not capable of much useful Avork. and when tbeir 
dependents usually have no insurable pecuniary interest in their lives. 



NOTES ON LIFE INSURANCE. 127 

Insurance coupled with endowment, payable at age 75, or at death if 
prior, would have no objectionable features if taken out at the 
younger ages, or even before age 50. But short-term insurance, 
coupled with endowment, is a specious delusion to those who do 
not look closely into its practical effect. In illustration of this, take 
a policy at age 30 to secure $10,000, to be paid to the policy-holder at 
age 40, in case he is alive at that time ; or to his heirs in case of his 
death if prior. This forms two distinct agreements in one contract. 
Assuming that the insurance clement of the policy is clearly under- 
stood in all its features, including the cost, we will consider the 
endowment element separately. The calculation of the net single 
premium to secure this endowment is made as follows (Actuaries' 
four per cent) : The amount that will produce $1 in one year 
at this rate is $0.96153846; multiply this quantity by itself nine 
times, or raise it to the tenth power, and we have $0.67556417, 
and this is the amount that will produce $1 in 10 years, at 
four per cent, compounded annually. But the amount is only to 
be paid in case the insured is alive at the end of the ten years. 
From the mortality table we are now using, we find that out of 86,- 
292 persons living at age 30, there will be 78,653 living at age 40 ; 
therefore, 78,653, divided by 86,292, expresses the fraction that at 
age 30 represents the probability that the person will be alive at age 
40. Hence, $0.67556417, multiplied by this fraction, gives the 
amount which, if paid in hand at age 30, will insure an endowment 
of 61 to be paid to the insured at age 40, if he is alive. Multiply 
this by 10,000, and we have $6157.60, which is the net single pre- 
mium for an endowment of $10,000 as above. This net single pre- 
mium, invested for ten years at 4 per cent compound interest, will 
produce $9114.75 certain; at 6 per cent, $11,027.32 ; at 10 per cent, 
$15,971.22. 

To provide for expenses, net premiums are increased. Suppose 
the loading in this case is 20 per cent; the actual premium paid in 
advance will then be $7389.12. This invested at 4 per cent com- 
pound interest amounts in 10 years to $10,937.70 ; at 6 per cent, 
$13,232.78 ; at 10 per cent, $19,165.47 ; and the endowment of $10,- 
000 is n6t to be paid to the insured unless he is alive at age 40. No 
prudent man who needs insurance should ever allow it to be coupled 
with short-term endowment. This remark applies in good degree 
to the " return premium plan," as it is called, by which the company 
obtains the use of a large amount of the policy-holder's money in 
excess of that needed to effect the insurance proper, and agrees to re- 
turn all the policy-holder has paid, without interest. Another ob- 
jection to this " return premium plan " is the comparative complexity 



12S NOTES ON LIFE INSURANCE. 

of the calculations. Joint-life insurance may be desirable, perhaps, 
in a few exceptional cases. The same may be said of survivor- 
ships. There is not much business of this kind done in the United 
States, and it will probably be well if the amount diminishes. 

Tontine Life Insurance. — T::e tontine principle gives certain speci- 
fied advantages to survivals at the cost of their a— 
instance, one hundred persons may put up one thousand dollars each 
in the hands of trustees ; the condition being that, at the end of 

twenty years, the whole fund and its accumulations shall be divided 
equally among the survivors, Tontines are of great variety in 
terms and conditions. Those policy-holders of a regular company 
who insure on this plan are placed in groups or classes by them- 
selves. All of each group who allow their policies to lapse before 
the end of the tontine period forfeit the accrued deposit and accu- 
mulations. Those who die in that time receive the amount of the 
policy without increase from over-pavnients. Xo return of over- 
payments is made to any holders of these tontine policies before the 
expiration of the period. This leaves until that time a large amount 
in the hands of the company that would in other kinds of insurance 
be returned yearly to the policy-holders. 

It is believed that the tontine principle had be::er no: be rnixc f. 
with life insurance proper. Whenever it is so mixed, the groups 
should at least be clearly defined, so that all may know, from time to 
time, just how the accounts in these groups stand. 

The Co-operative Flan. — This system or scheme is based upon 
trance fees to pay expenses, and voluntary contributions after death 
has occurred, to pay losses. Considered as benevolent and chari- 
table institutions amongst certain professions or brotherhoods, these 
co-operative associations may, under certain circumstances, be of great 
benefit to a few individuals. But there is no business basis in it, 
and not a single feature that entitles such an organization to be called 
an insurance company. They, however, often assume the name and 
claim to furnish insurance with more certainty and at less ■:■: c : than 
can be done by the largest and best conducted purely mutual com- 
pany that demands money in advance before contracting to insure 
lives and return all surplus. 

These co-operatives usually die out soon, but new ones spring up, 
and this will no doubt continue until business men find out that to 
secure real life insurance it must he paid for in advance. As a : ... 
it will not do, in the long run, to trust this matter t ibutions to 

be made, if at all, sixty days after the policy-holder is dead. 



NOTES ON LIFE INSURANCE. 129 



CHAPTER XL 

GROSS VALUATIONS— NET VALUATIONS. 

By the gross method of valuing life policies, it is assumed that the 
future expenses will be less than the loading ; and that, after making 
a reasonable estimate for these expenses, the remainder of the load- 
ing may be considered as profit, and the present value thereof en- 
tered in the assets of the company. The expenses may, for pur- 
poses of illustration, be fairly estimated at 15 per cent, and the 
loading at 30 per cent of the net premium. 

In case the net annual premium at age 30 for a whole-life policy 
is $100, and the loading is $30, the expenses being $15 each year, 
there will then be $15 paid each year in excess of what is required 
to effect the insurance and pay expenses. Calculate the value at 
age 30 of a life series of annual payments of $15 each. To do this, 
divide N at age 30 by D at the same age. This will give, by the 
American Experience Table of Mortality, and 4|- per cent interest, 
$17.12 as the value at age 30 of a life series of payments of $1 each. 
Multiply this by 15, and we have the value at age 30 of a life series 
of annual payments of $15 each. This value is $256.80. By con- 
sidering this as a realized asset, the accounts would seem to show 
that the company in making this contract had immediately become 
$256.80 richer by this operation; because it has received sufficient 
to effect the insurance it has contracted to furnish, pay expenses, 
and, in addition, is to receive $15 a year for life from the insured. 

Thirty thousand such policies as the above would, by this method 

of gross valuation, result in entering at once among the assets $7,- 

704,000 clear profit. This too, notwithstanding the fact that the 

whole sum received by the company is but $3,900,000 out of which 

^ to pay death losses and expenses during the year, and provide for 

, the deposit that must be held at the end of the year. 

In this connection, attention is called to the following extract 
from the report of a committee of the British Parliament in 1853. 
(The party under examination was the actuary of the " Royal Ex- 
change Assurance Office.") 



130 NOTES OX LIFE INSURANCE. 

Question : " Do you think there is any thing peculiar in the cha- 
racter of life assurance business which would justify the legisla- 
ture in interfering with it in a way different from other business ?" 

Answer; ;; Yes ; both on account of the long period over which 
the contracts extend, and especially for this reason : that life assur- 
ance offices are now taking to make up their accounts on principles 
that would be scouted from any other department of commercial 
enterprise." 

Question : ' ; Will you explain what principle you mean ?" 

Answer: '"'The practice of anticipating future profits and treating 
them as assets. Allow me to suppose the case of a bank making up 
its accounts : it owes to its depositors £1.000,000 ; it has on hand 
£900,000 ; it puts down as an additional item of assets, profits, we 
will say. at the rate of £10,000 a year, valued at twenty years' pur- 
chase ; by that means, it makes its assets £1,100,000 against £1,000,- 
000, and the result is stated to be a surplus of £100,000. That 
principle would never be adopted in a bank, and I think it ought not 
to be adopted in an assurance company/' 

Question: "But does it exist in assurance companies?" 

Answer : u It is done/' 

Question : " Is it done by assurance companies generally, or only 
in particular cases ':" 

Answer: ;; It is in considerable use, and the practice is extend- 
ing." 

It is a safe business rule not to estimate your present wealth and 
regulate your present expenses by what you suppose your clear pro- 
fits will be in future years. "When the first payment of $15 in ex- 
cess of the net premium and expenses has been realized by the com- 
pany out of the premiums paid by each of the thirty thousand policy- 
holders, the profits from this source, namely, $450,000, may be en- 
tered in the assets. The remainder of the $7,704,000 should not be 
counted as assets before it is realized. 

In case the " loading " and other resources of a company should 
prove to be in excess of expenses, and all claims other than those 
provided for by net premiums and net interest thereon, profit will 
result. But in view of the great and increasing number of policies 
that lapse or are surrendered, it would be dangerous to permit com- 
panies to assume that all their existing policies will continue in the 
company to maturity, and that their future yearly expenses will be 
certainly a given amount less than the loading, and place in their 
accounts, under the guise of assets realized and on hand, the amount 
of supposed profits the company may make in the future. 



NOTES ON LIFE INSURANCE. 131 

Lapsed Policies. — Policies are often allowed to lapse from inabili- 
ty to pay the premium. Sometimes this occurs because the policy- 
holder no longer needs to insure his life. But it is believed that 
much the larger portion of surrendered and lapsed policies arise from 
misapprehension on the part of policy-holders at the time of taking 
out the policy in regard to its precise nature and effect. It is not 
harsh to say that this arises often from the fact that agents do not 
take the pains to explain, even when they themselves understand, the 
exact nature of the policy they sell. 

The companies have not, as a rule, shown any over-anxiety to have 
other than favorable views presented to the policy-holder at the 
time of signing the contract. 

Full information, fair and candid dealing at the time the policy is 
issued is absolutely essential, if companies desire to diminish the 
number of polices surrendered and allowed to lapse. 

This general principle is applicable to all kinds of business. Life 
insurance forms no exception to this rule, nor to the fact that suc- 
cess in business does not necessarily depend upon the amount of 
business done. The terms of the contract — the policy — should be 
made explicit and fair. 

Knowledge of life insurance, full- and exact information, is what 
is needed. It will not hurt officers of companies, directors, trustees, 
or agents, and it is essential that some of the policy-holders should 
understand this subject. 

The Legal Standard of Safety. — " Something more than bare com- 
mercial solvency is required of life insurance companies." The laws 
of many of the States are intended to guard with especial care these 
trust funds held by corporations for future widows and orphans. 
The character of the securities in which these funds may be invested 
is prescribed ; a deposit of one hundred thousand dollars with a 
principal financial officer of the State is required ; the law determines 
the various items that may be admitted as assets, and designates 
the table of mortality and rate of interest to be used as a basis for 
calculating the liability of the company on account of the deposit 
that must be held for all policies the company has in force. On this 
point, the law in one or more of the States provides that — 

" When the actual funds of any life insurance company doing 
business in this Commonwealth are not of a net cash value equal to 
its liabilities, counting as such the net value of its policies, accord- 
ing to the * American Experience ' rate of mortality, with interest 
at four and one half per centum per annum, it shall be the duty of 
the Insurance Commissioner to give notice to such company and its 



132 NOTES ON LIFE INSURANCE. 

agents to discontinue issuing new policies within this Common- 
wealth until such time as its funds have become equal to its liabili- 
ties, valuing its policies as aforesaid." 

The question of commercial solvency is not raised by carrying 
into effect the foregoing requirements of law. But it is made the 
duty of the commissioner, after giving notice to the company to 
cease issuing new policies, to examine all its affairs, and if he is " of 
opinion," after such examination, that " a company is insolvent, or 
that its condition is such as to render its further proceedings hazard- 
ous to the public, or to those holding its policies, he shall report to 
the Attorney-General, who shall bring the matter before a court of 
competent jurisdiction; and the court, after full hearing, shall 
make such orders and decrees as may be needful, according to the 
usual course of proceedings in equity." 

In strictly adhering to the system of net valuation as the legal 
standard of safety for a life insurance company, and requiring that 
the admitted assets of such a company shall be held in the securi- 
ties prescribed by law, it does not follow that when a company 
fails to come up to this standard it should necessarily be given 
over to be divided in pieces, and its funds absorbed in the never- 
ending expenses of a chancery court. 

The failure of a company to stand the test of net valuation com- 
puted upon data designated by the State, is an alarm-bell that 
should be heeded by all. In case it is, on close examination of all 
the facts, clear that the difficulty can not be remedied, then the 
assets of the company should be equitably and promptly distributed 
to the legal owners thereof. In case it is clear that a company may 
recuperate, by reducing expenses below the loading, and giving 
time for improvement in the condition of its affairs, arising from a 
rate of mortality that may be less, and a rate of interest greater 
than that upon which the net premiums are based, it will be well to 
allow the company time to endeavor to retrieve its lost ground, and 
re-establish its impaired deposit. But this should only be done 
when there is good reason to believe that the interests of the policy- 
holders will be benefited thereby, and not for the purpose of per- 
mitting companies to continue in business, merely because it is as- 
sumed by them that they will, in the future, make great profits. 

In at least one State the law requires that a life insurance com- 
pany shall cease to issue new policies whenever the admitted assets 
are twenty thousand dollars less than its liabilities, including in the 
latter the one hundred thousand dollars deposited with a principal 
financial officer of the State. It would seem that this should be the 



NOTES ON LIFE INSURANCE. 133 

law iii all the States, because there is an evident absurdity in requir- 
ing this amount to be deposited with the treasurer or other State 
officer for the better security of all the policy-holders of the com- 
pany in the United States, and then permitting the company to at 
once incur liabilities for the whole of this amount. 

But existing laws in nearly all of the States authorize life insur- 
ance companies to continue issuing new policies until the whole 
capital, including the one hundred thousand dollars deposited with 
the State, is exhausted. 

Net Valuation. — The " loading" upon net premiums is, as previous- 
ly stated, intended to provide for expenses, promts, and adverse 
contingencies. The State, in prescribing a table of mortality and 
rate of interest upon which to calculate net values in life insurance, 
designates the amount that will on these data safely effect the insur- 
ance. In calculating the deposit or reserve that must be held by a 
company at any time to the credit of a policy, the law authorizes the 
value of the future net premiums to be deducted from the net single 
premium. This is done because these net premiums together are, 
upon the data originally assumed, just sufficient to effect the insurance. 
If the insured at any future time fails to pay his net premiums, the 
insurance ceases, and no harm results from having at a previous time 
credited the company with the value of these future net premiums. 
This does not, however, hold good in case a company has been 
credited with that part of the loading on these future net pre- 
miums, not required for expenses, simply because in failing to 
receive this part of the loading, the company is not relieved from 
any equivalent obligation, and this item, previously admitted as a 
valid credit, falls to zero in case future premiums are not paid. 
And as this contingency may happen with respect to any policy — 
and often does occur — the State has fixed upon the method of net 
valuation as the standard of safety. 

Some companies that base their net premiums on six per cent, 
maintain that they fully comply with the four and a half per cent 
standard, when they have a deposit for each policy equal to that 
called for by the valuation tables computed upon the four and a 
half per cent basis. In this they ignore the fact that these tables 
assume that the future net premiums the company has contracted 
to receive are those called for by the State standard of safety. 
Therefore, the results given in these tables are not correct, in case 
the company has contracted to furnish insurance for future net 
premiums less than those called for by the State standard. 



134 



NOTES ON LIFE INSTJKANCE. 



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NOTES ON LIFE INSURANCE. 137 

lations, the company will not only get back the original deposit with 
interest thereon, but it will make a clear profit on the policy that 
persists to the table limit, &3604.6G, besides paying the policy of 
$1000 at age 96. 

It will be noticed that the table is constructed npon net values, 
and it should be borne in mind that the net premiums are in practice 
loaded in order to provided for expenses and contingencies ; that the 
loading is generally more than enough to pay expenses, and that 
when a limited number of shareholders receive profits yearly, arising 
from the excess of loading over expenses, on a large number of pre- 
miums, the profits from this source alone may well be very great. 

It is maintained by those who are believed to know, that the 
tables of mortality in use show a higher death-rate than will actually 
occur amongst well-selected lives, which, if true, will be a large ad- 
ditional source of profit to the shareholders, besides profits that may 
and do often arise from surrender charges and forfeitures. 

It is claimed by some that human lives of the same age can be 
classed in such a manner that the long-lived may be assured at one 
price, and the short-lived at another and greater price, a good deal 
in the manner in which different kinds of property are classified in 
fire insurance. In short, they insure certain classes at a higher and 
others at a lower rate than the medium or average risk. It has, how- 
ever, not yet been found generally practicable and safe to regulate 
the net price of life insurance on the assumed individual longevity 
of each isolated policy-holder. 

The representatives of some companies insist that it is not just or 
reasonable to apply to them the test prescribed by the State, of net 
values based upon a designated table of mortality and rate of inte- 
rest, because, they say, they have selected extra good risks, and that 
the death-rate amongst the insured in their companies will certainly 
be less than that given by the table of mortality. If their esti- 
mates prove to be correct in this respect, the company will eventu- 
ally reap the benefit arising from a lower death-rate amongst its 
policy-holders. But the State assumes that it is safer for all compa- 
nies to be held, at the end of each year, to a designated general 
standard based noon observation and experience. 



13S NOTES ON LIFE INSURANCE. 



CHAPTER XII. 

THE DEPOSIT WHEN A RENEWAL PREMIUM IS NOT PAID SHOULD BE 
USED FOR EFFECTING FULL-PAID INSURANCE— ANNUAL STATE- 
MENTS. 

Iisr case a life policy is not renewed by the payment of the annual 
premium when due, the company has in its hands the deposit in- 
tended solely for the future insurance of this policy. The policy- 
holder has paid for all the insurance the company has previously 
furnished him ; paid his share of the previous expenses ; contribut- 
ed his proportion toward the profits, and the company holds a 
deposit paid by him for future insurance. It is claimed by some 
that the withdrawal of a policy-holder is an injury to those who 
remain in the company, and that on the non-payment of a renewal 
premium, he should forfeit the deposit held by the company. 

The justice of this forfeiture is denied. So long as the death 
rate among the insured conforms to that given in the table of mor- 
tality, and the interest realized is that assumed, the net premiums 
are just sufficient to pay death losses year by year, and provide the 
requisite deposit for each policy. 

Therefore, so far as paying death losses is concerned, and con- 
sidering the net premiums only, it makes no difference to the com- 
pany whether it has a greater or less number of policies in force, 
provided the mortality conforms to the rate designated in the table. 
The number insured may be 100,000 or 1,000,000 ; the company with 
one million of policy-holders would make no more out of the normal 
net contributions of each policy-holder to pay death-claims than the 
company with one hundred thousand policy-holders would make out 
of the net premiums of each of its policy-holders. 

In other words, in case 900,000 policy-holders withdraw from a 
company containing 1,000,000, so far as net values are concerned, 
the company, when reduced to 100,000 policy-holders, is just exactly 
as strong in resources for paying death -claims as it was before the 
900,000 withdrew. This is so, because when nine tenths of them 
withdraw, and diminish the net premiums of the company 90 per 
cent, their withdrawal diminishes the death-claims iu just the same 



NOTES ON LIFE INSURANCE. 139 

proportion. Expense incurred in the transaction of the business of 
a company is a different thing. The working expenses of a com- 
pany consisting of 100,000 policy-holders ought to be less on each 
policy than that of a company consisting of 1000 policy-holders 
should be on each of its policies. 

In estimating "the value of a policy-holder to a company to keep," 
we pass from the theory of net values to the consideration of ordi- 
nary business matters, such as expenses, loading, interest realized in 
excess of table rate, the health of the policy-holder at the time of 
proposed withdrawal, the cost of getting a policy-holder into the 
company, and whatever may have a practical bearing on the par- 
ticular case. In determining what should be done with the accrued 
deposit in case the contract for insurance is not renewed at the be- 
ginning of any policy year, it should not be forgotten that the in- 
surance the company furnishes is not the amount called for by the 
policy — it is the amount at risk, which is the face of the policy, less 
this deposit. 

Until a very recent period, it was the custom of life insurance 
companies to appropriate to themselves the whole of the accrued 
deposit, in case the insured failed to renew the contract by paying 
the annual premium when due. The rights of withdrawing policy- 
holders in this regard are getting to be somewhat better understood, 
and many of the companies show a willingness, at least in part, to 
respect them. 

The insured has no right, in equity, at his own option, to demand, 
at the time of withdrawal, the return to him of any portion of the 
deposit accrued on his policy. The original contract was for in- 
surance ; the deposit is intended to provide for future insurance; and 
all that the policy-holder is in equity entitled to is the insurance that 
this deposit will at that time effect. 

In determining this amount, the future expenses of the policy must 
be provided for ; and the remainder of the deposit only can be pro- 
perly used as a net single premium for full-paid insurance. No 
further collections of premiums have to be made, and no agent's 
commissions thereon paid ; therefore the expense will probably be 
little else than that attendant upon the investment and keeping of 
the funds. The interest actually received being usually more than 
the table net rate, this excess of interest might well be more than 
enough to cover the necessary expense upon the policy. When the 
accrued deposit is large, and the amount at risk consequently small 
compared with the policy, it seems absurd, as authorized by law in 



140 NOTES ON LIFE INSURANCE. 

some States, to allow the company to deduct 20 per cent of the de- 
posit, and furnish full-paid insurance for the remainder only. 

This, however, is greatly better for the withdrawing policy-holders 
than the former system under which the company appropriated to 
itself the whole of the deposit in case of the non-payment of a re- 
newal premium when due. If those who procured the passage of 
the State laws, just referred to, had based the deductions on the 
amount at risk, instead of on the amount of the deposit, it would 
have been more equitable. 

Take, for instance, the deduction of 20 per cent of the deposit at 
the end of the first year ; find what per cent this sum is of the 
amount at risk the first year. Having thus fixed the rate per cent 
of the amount at risk to be used in computing the sum to be de- 
ducted from the deposit, the deduction will be less as the policy 
grows older, instead of increasing as it does under existing laws, 
above referred to. For example, the deposit at the end of the first 
year for an ordinary life policy of 810,000, issued at age 20 (Am. Ex. 
4-J-) is 847.36. Therefore the amount at risk the first year is 
$9952.64. 

Twenty per cent of the deposit at the end of the first year is 
89.472. This sum is equivalent to 10 g^ 00 (which is a small fraction 
less than one tenth of one per cent) of the amount at risk the first 
year. At the end of the first year, then, the deduction of twenty 
per cent from the deposit gives substantially the same as a deduc- 
tion of one tenth of one per cent of the amount at risk. At the 
end of the fiftieth policy year, the deposit is 86235.96. Therefore 
the amount at risk that year is 83764.04. The 20 per cent de- 
ducted before applying this deposit to the purchase of full-paid 
insurance is 81247.19. One tenth of one per cent of the amount at 
risk is $3.76. 

At age 94, the deposit is 89449.72. The amount at risk during 
the year is 8550.28. Twenty per cent, to be deducted before apply- 
ing this deposit to the purchase of full-paid insurance, is 81889.94. 
One tenth of one per cent of the amount at risk is 80.55. 

The glaring inequity of a rule that allows a deduction of 81889.94 
to be made from the deposit when the whole amount at risk during 
the year is but $550.28 needs no comment. 

It is maintained by some that the policy-holders who cease to 
pay renewal premiums when due are nearly always in sound health, 
with fair prospects for long life, and that the unhealthy policy- 
holders remain. This selection against the company will, it is 
claimed, result in a very high rate of mortality amongst those that 
continue. 



NOTES ON LIFE INSURANCE. 141 

It is not at all clear that any large proportion of policy-holders 
cease to pay renewal premiums merely because they are in good 
health. But, be this as it may, existing legal contracts must stand 
unless modified with the consent of all the parties ; but in contracts 
yet to be made the companies should not be permitted to seize upon 
and appropriate to themselves the accrued net value of life in- 
surance policies. After making a deduction therefrom for future 
expenses, and fair compensation for diminished vitality in the com- 
pany, caused by the failure to pay a renewal premium, the remainder 
should be used as a net single premium for full-paid insurance — 
the term of this full-paid insurance being that named in the original 
contract, the amount being determined by the net single premium. 

ANNUAL STATEMENTS. 

In many of the States, all the companies that do business therein 
are required by law to make annual statements to the insurance 
commissioner. 

These statements give the assets and liabilities at the end of the 
year, and income and expenditures during the year, in full detail ; 
and in addition a balance-sheet, taking as a basis the assets of the 
company at the beginning of the year. 

The balance-sheet is an effective probe, often resisted by those 
who do not care to fully expose all the details of the business of a 
company, or take the trouble to explain them. It is believed that 
the following explanation will illustrate to policy-holders the impor- 
tance of having this requirement of the law enforced upon those 
who hold these trust funds. 

In the balance-sheet, for the end of any year, the assets on hand 
at the beginning of the year are taken as the basis ; add to this the 
income during the year, and gains, if any, in market value of secu- 
rities and other property ; gains from accrued interest ; gains in 
amount of uncollected premiums, in rents, and in any other item not 
included, either in assets at the beginning of the year or in income 
during the year. This gives the amount to be accounted for at the 
end of the year. From this deduct the expenditures — deduct the 
depreciation, if any, during the year, in market value of securities 
and other property — and deduct all other losses or depreciations in 
market value. When these deductions are made from the amount 
to be accounted for, the result gives the market value of the assets 
that ought to be on hand. If the required assets are on hand, the 
account is correct ; but if the assets on hand at the end of the year 
are either more or less than the amount called for after the specified 



142 NOTES ON LIFE INSURANCE. 

deductions are made, there is an error which should be found and 
corrected. The details of gains and depreciations are required to 
be given in explanatory schedules, the form of which each company 
arranges to suit its own business and books. In illustration, sup- 
2>ose the result as given by a detailed schedule of the investment ac- 
count made out by a company shows a gain of $10,000. This would 
be added to the assets at the beginning of the year, and indicated 
in the balance-sheet under the head of u Balance of investment ac- 
count, credit side ;" but if the detailed schedule of the investment 
account showed a loss during the year of $10,000, this in the ba- 
lance-sheet would be added to the expenditures, and indicated by 
" Balance of investment account, debit side." The blank f orm of 
balance-sheet provides, on both the credit and debit sides, for " ba- 
lance of investment account ;" so that if the balance,- as shown by 
the schedule, is gai?i i it may be added to the assets on hand at the 
beginning of the year ; or if the balance is loss, it may be added to 
the expenditures. The same applies to the schedule of profit and 
loss, showing this detailed account. If this account shows a gam 
of $10,000, the form of balance-sheet provides under the heading, 
u Balance of profit and loss account, credit side," for adding this to 
the assets on hand at the beginning of the year ; but if this detailed 
schedule shows a loss of $10,000, the blank form of balance-sheet 
provides that this be added to the expenditures, and it is entered on 
that side of the balance-sheet, under the heading, " Balance of pro- 
fit and loss account, debit side." 

For instance, suppose the assets at the beginning of the year 
amount to $500,000, the cash income is $1,000,000 ; gains in market 
value and other gains, $100,000. This gives $1,600,000 to be ac- 
counted for. Suppose the expenditures to have been $700,000. 
This indicates that there should be $900,000 assets at the end of the 
year. Suppose the assets are only worth $800,000, the balance- 
sheet and its explanatory schedules must account, item by item, for 
the missing $100,000. 

These annual statements are required to be made in the form that 
may be prescribed by the commissioner, and give all the information 
nsked for by him in regard to the business and affairs of the com- 
pany, the whole verified by the signature and oath of the president 
and secretary of the company. 

The law further provides that whenever the commissioner sus- 
pects the correctness of any annual statement, or that the affairs of 
any company making such statement are in an unsound condition, 
he shall visit and examine such company. At such times, he shall 



NOTES ON LIFE INSURANCE. 143 

have access to its books and papers, and shall thoroughly inspect 
and examine all its affairs, and he may summon and examine, under 
oath, the directors, officers, and agents of any insurance company, 
and such other persons as he may think proper, in relation to the 
affairs, transactions, and condition of said company. 

The law in many of the States has imposed the foregoing and 
other important conditions upon life insurance companies ; but, after 
all, a great deal is left to the judgment and discretion of the policy- 
holder in selecting the kind of insurance he needs, and the company 
in whose hands he proposes to place funds in trust for the benefit of 
his heirs. No person should make an application to a company for 
insurance on his life until he has carefully read the form of the po- 
licy which is to be the contract between himself and the company. 

Life insurance will bear the closest scrutiny. It needs it. The 
most important element in the accounts is an accurate registry, or 
descriptive list of the policies in force. Without this, no computa- 
tion can be made of the accrued liability of a company, on account 
of the net value of its policies, which is the amount the company 
should have on hand deposited to the credit of the policies. 

u In some cases, life policies have been reported not in force, the 
" company thereby escaping liability for the net value thereof, when, 
" in fact, the terms of these policies obligated the company for a spe- 
" cific amount of full-paid insurance in lieu of the policy reported 
" not in force." After getting an exact description of all the poli- 
cies in force, and calculating the liability of the company on ac- 
count of the accrued net value of these policies, and adding thereto 
all other liabilities of the company, it becomes important to know 
the nature and amount of assets held by the company to meet its 
liabilities. Even mortgages, on improved, productive, unencumber- 
ed real estate, worth more than double the amount loaned thereon, 
have been made delusive. 

Well, might Mr. Gladstone, in a speech delivered March 7th, 
1864, in the British House of Commons, state, " I dare say there are 
4< not many members of this house who know to what an extraordi- 
" nary extent the public are dependent on the prudence, the high 
" honor, and the character of those concerned in the management of 
" these institutions." 



APPENDIX. 



APPENDIX 



CHAPTER XIII. 

EXTRACTS FROM MASSERES ON ANNUITIES, LONDON, 1783, AND QUO- 
TATIONS FROM AMERICAN ACTUARIES, 1870, 1853, AND 1871. 

These extracts show an earnest desire on the part of Masseres to 
convey to his readers a thorough understanding of the principles on 
which calculations of the " True Value" or " Fair Price" of annui- 
ties are based. 

Extracts — Masseres, 1783. 

An explanation of the data or grounds upon which the computa- 
tions of the values of annuities for lives are built. These are, first, 
the decrease of the present value of a future sum of money arising 
from the mere distance of time at which it is to be paid, and the 
consequent discount that is to be allowed to the purchaser of it for 
prompt payment (the quantity of which discount, it is evident, will 
depend on the rate of interest of money) ; and, secondly, the chance 
which, when the payment of such future sum is not made certain, 
but is to depend on the continuance of the life of a person of a 
given age, the grantor of it has of escaping the necessity of paying 
it at all by means of the death of the said person before it becomes 
due, in order to determine which chance, it is necessary to have re- 
course to certain tables of the several probabilities of the duration 
of human life at every different year of age, which have been formed 
from observations of the numbers of persons who have died every 
year, in the course of a long series of years, at different ages, in di- 
vers cities and parishes, and other numerous bodies of men. 

The doctrine of life annuities is by no means of so abstruse and 
difficult a nature as many people are apt to imagine. A moderate 
share of common sense, or capacity to reason justly, and a know- 
ledge of common arithmetic, are all the qualities that are necessary 
to a right understanding of the principles on which it is founded, 
even so far as to be able to compute the value of any proposed an- 



148 



NOTES ON LIFE INSURANCE. 



nuity for any given life or number of lives, if a person is disposed 
to undergo the labor of performing all the necessary arithmetical 
operations that arise in such a computation. 



TABLE 



Representing the probabilities of the duration of human life at the 
several ages therein mentioned, from the age of three years to the 
age of 95, grounded on lists of the French Tontixes or Long Axxui- 
ties, and verified by a comparison thereof with the necrologies, or 
mortuary registers, of several religious houses of both sexes. 



BY MONSIEUR DE PABCIEEX. 



Age. 


Persons 


Age. 


Persons 


Age. 


Persons 


Age. 


Persons 


Age. 


Persons 


Living. 


Living. 


Living. 


Living. 


Living. 


3 


1000 


22 


798 


41 


650 


60 


463 


| n 


136 


4 


970 


23 


790 


42 


643 


61 


450 


80 


118 


5 


943 


24 


782 


43 


636 


62 


437 


1 81 


101 


G 


930 


25 


774 


44 


629 


63 


423 


82 


85 


7 


915 


26 


766 


45 


622 


64 


409 


: 83 


71 


8 


902 


27 


758 


46 


615 


65 


395 


84 


59 


9 


890 


28 


750 


47 


607 


66 


380 


85 


48 


30 


880 


29 


742 


48 


599 


67 


364 


86 


38 


11 


872 


30 


734 


49 


590 


68 


347 


87 


29 


12 


866 


31 


726 


50 


581 


69 


329. 


! 88 


22 


13 


860 


32 


718 


51 


571 


70 


310 


i 89 


16 


14 


854 


33 


710 


52 


560 


71 


25)1 


90 


11 


15 


848 


34 


702 


53 


549 


72 


271 


91 


7 


16 


842 


35 


694 


54 


538 


73 


251 


| 92 


4 


17 


835 


36 


686 


55 


526 


74 


231 


93 


2 


18 


828 


37 


678 


56 


514 


75 


211 


! 94 


1 


19 


821 


38 


671 


57 


502 


76 


192 


95 





20 


814 


39 


664 


58 


489 


. 77 


173 






21 


8.6 


40 


657 


59 


476 


78 


154 







The Fundamental Maxim of the Doctrine of Life Annuities. — 
In every bargain between two persons concerning a grant of a sum 
of money to be paid by the one to the other at a given future time, 
in case the grantee or purchaser shall be then alive, or in case the 
grantee and one or more other persons of given ages shall be then 
alive, the fair price of such future sum of money, according to a 
given rate of the interest of money and a given table of the pro- 
babilities of the duration of human life, is to be ascertained in the 
following manner: We must suppose, in the first place, that the 
grantor of the future sum of money makes several hundred grants 
of the same kind, and upon exactly the same conditions, to as many 
different grantees, or purchasers, all of the same age with the first 
grantee ; and, in the second place, that these several purchasers and 
their companions (or the persons upon the continuance of whose lives, 
as well as their own, their right to the said future sums depends) die 
off in the interval between the time of making the grants and the 



NOTES ON LIFE INSURANCE. 149 

time of payment, in the same proportion as persons of the same 
ages respectively are represented to do in the table of the probabi- 
lities of the duration of human life by which the calculation is to be 
governed ; and, in the third place, we must suppose that the several 
sums of money paid by the several grantees of these future pay- 
ments to the grantor of them as the price thereof, are improved by 
the said grantor, at compound interest, at the rate supposed in the 
question, during the whole interval of time between the time of 
making the grants and the time at which the payments become due. 
And then we must inquire what sum each of the said grantees ought 
to pay to the grantor, to the end that, upon these three suppositions, 
he may, at the end of the said interval, or when the said payments 
become due, be neither a gainer nor a loser by the sum total of all 
his bargains, but be possessed of just enough money, arising from 
the sums formerly paid him by the said grantees, to satisfy all the 
demands which will then be made upon him. And the sum which 
ought thus to be paid him by each of the said grantees, when he 
makes a great number of said grants to different persons, is the fair 
price which a single grantee ought to pay him for a grant for the 
said future sum of money, subject to the same conditions and con- 
tingencies, when he makes only one such grant. 

This is a maxim which, I presume, will be admitted as self-evi- 
dent, it being hardly possible to doubt of its truth. But if the 
reader should not admit it upon its own evidence, I confess I am 
unable to demonstrate it by means of any other proposition more 
evident than itself. And, therefore, in this case, I must desire him 
to consider it as a definition of what is meant in the following 
pages by the expressions of the fair price or true value of such a 
future contingent payment, since it is only in that sense that the 
fair price or true value of such a future contingent payment can be 
collected from the tables of the probabilities of the duration of hu- 
man life above described. 

The first problem is, " To find the present value of a future sum 
of money, which is certainly to be paid at the end of one or more 
years, according to any given rate of interest." 

PROBLEM I. 

To find the present value of any given sum of money which is 
payable at the end of any given number of years, according to any 
given rate of interest. 

Solution. (Omitted.) 



150 NOTES ON LIFE INSURANCE. 



PROBLEM II. 



To find the sum of money which the purchaser of a future pay- 
ment of one pound sterling, to be received at the end of any given 
number of years,, provided the said purchaser shall then be 'living, 
ought to pay for it— the age of the said purchaser, and the rate of 
interest of money, and the probabilities of the duration of human 
life, being all given. 

A Solution of this Problem in the Case of a Particular Example. 

Let the rate of interest of money be supposed to be 3 per cent, 
and the probabilities of the duration of human life such as they are 
represented to be in Monsieur de Parcieux's table above mentioned ; 
and let the number of years at the end of which the said sum of 
one pound is to be paid to the grantee, or purchaser of it, if he be 
then alive, be 30, and the age of the said grantee, or purchaser, 25 
years. 

Then, in the first place, we must look into M. de Parcieux's table 
to see how many persons of 25 years of age are there supposed to 
be all living at the same time. This number we shall find to be 774. 
We must therefore suppose that the grantor of the one pound to 
the purchaser, proposed in the question, does not confine himself to 
that single grant, but makes 773 more such grants, of one pound 
each, to as many different persons of the same age of 25 years, to be 
paid to them at the end of 30 years, or when they shall be 55 years 
old, if they shall then be living, but not to be paid to their execu- 
tors, or other representatives, if they shall then be dead ; that is, we 
must suppose that he makes 774 such grants in all, including that of 
the purchaser proposed in the question. And we must likewise 
suppose that all these 774 purchasers have the same chance, one 
with the other, of living any given number of years, or that there 
is no apparent reason for supposing that any one of them is more 
likely to live to any given future age than any other. This done, 
we must inquire how many of these 774 purchasers of one pound 
each will be alive at the end of 30 years, supposing them to die off 
in the proportion mentioned in M. de Parcieux's table. ISTow, it ap- 
pears by M. de Parcieux's table, that out of 774 persons of the age 
of 25 years, all living at the same time, 526 will be alive at the age 
of 55 years, or at the distance of 30 years. Therefore, out of the 
said 774 purchasers of these future payments of one pound, to be 
received at the end of 30 years, 526 will live to be entitled to them. 



NOTES ON LIFE INSURANCE. 151 

Therefore at the end of the said 30 years, the grantor of these 
future payments will have 52G sums, of one pound each, to pay to 
the said surviving purchasers. And consequently, to the end that 
the said grantor may be neither a gainer nor a loser by the sum 
total of all his bargains, it is necessary that he should receive at the 
time of making the said grants 526 times the present value of one 
pound, payable at the end of 30 years, when the interest of money 
is 3 per cent, or 526 times the sum which, being improved continu- 
ally at compound interest during the said term of 30 years at the 
said rate of interest, will at the end of that time amount to one 
pound; because, in that case, if he improves the said sum (of 526 
times the present value of one pound) so received, at compound in- 
terest, at the said rate of 3 per cent, during the whole 30 years, it 
will in that time increase to just 526 pounds, which is the sum he 
will then be obliged to pay to the surviving purchasers. The pre- 
sent value of one pound, payable at the end of 30 years, without 
being liable to any contingency, when the interest of money is 3 
per cent, is .41198676 of a pound. Therefore 526 times .41198676 
of a pound, or £216.70503576, is the sum which the said grantor 
ought to receive, at the time of making the said grants, from all the 
774 purchasers of them. Therefore, the sum which each of them 
ought then to pay him is the 774th part of £216.70503576, or 
.27998066 of a pound, or nearly .28 of a pound, or 55. I^d. And 
consequently when he makes only one such grant to a purchaser of 
25 years of age, he ought to receive for it the same sum of 
.27998066 of a pound, or .28 of a pound, or 5s. *l\d. 

I have solved the foregoing problem, in the case of a particular 
example, for the sake of making the method of solution as clear and 
familiar as possible. But it is easy to see that the reasonings used 
in it extend to all other cases whatsoever, and consequently that the 
solution is really general. 

Problem 3d relates to the computation of the present value of a 
future sum of one pound sterling, that is to take place at the end of 
a certain number of years, provided two persons of given ages shall 
then be living, and upon the supposition of a given rate of interest 
of money. 

PROBLEM III. 

To find the sum of money which the purchaser of a future pay- 
ment of one pound sterling, to be received at the end of any given 
number of years, provided the said purchaser and another person 
(who may be called his companion) shall be then living, ought to 



152 ROTES OX LIFE ENSFBAKGK 

pay for such future sum ; the ages of the said purchaser and his 
companion, and the rate of interest of money, and the probabilities 
of the duration of human life, being all given 

A Solution of this Problem in the Case of a Particular JSzample. 

Let the rate of interest of money be supposed to be 3 per cent, 
and the probabilities of the duration of human life to be such as 
they are represented to be in M. de Parcieux's table above men- 
tioned, and let the number of years at the end of which the said 
sum of one pound is to be paid to the purchaser of it, in case not 
only the said purchaser himself, but likewise his companion afore- 
said, shall be then alive, be 30 ; and the age of the said purchaser 
be 25 years ; and that of his said companion be 20 years. 

Then, in the first place, we must look into M. de Parcieux's table 
to see how many persons of 25 years of age are there represented as 
all living at the same time. This number is 774. We must there- 
fore suppose that the grantor of the one pound to the purchaser 
proposed in the question makes at the same time 773 more such 
grants of one pound to as many different persons all of the same 
age of 25 years, to be paid to them at the end of 30 years, or when 
they shall be 55 years old, if not only the grantees themselves shall 
then be living, but certain other persons, who may be called their 
companion^ who are of the same age of 20 years with the com- 
panion of the purchaser mentioned in the question ; that is, we must 
suppose that the grantor makes 774 such grants in all, including 
that of the purchaser proposed in the question. And we must like- 
wise suppose that all these 774 purchasers of these future payments 
of one pound have the same chance, one with another, of living 
any given number of years, or that there is no apparent reason for 
supposing that any one of them is more likely to live to any given 
future age than any other. This done, we must inquire how many 
of these 774 purchasers of these remote payments will be alive at 
the end of 30 years, supposing them to die off in that interval of 
time in the proportion mentioned in M. de Parcieux's table. Xow, 
it appears by M. de Parcieux's table, that out of 774 persons of the 
age of 25 years, all living at the same time, 526 will be alive at the 
age of 55 years, or at the end of 30 years. Therefore out of the said 
7 74 purchasers of these future payments of one pound each, only 
526 will live to the end of the 30 years. And of these 526 surviving 
purchasers, only some part will be entitled to demand these pay- 
ments — to wit, those whose companions, who were of the age of 20 
years at the time of making the grants, are likewise living at the 



NOTES ON" LIFE INSURANCE. 153 

end of 30 years. For the other surviving purchasers, whose com- 
panions are then dead, will, by the conditions of this problem, have 
no right to them. We must therefore, in the next place, inquire 
how many of the companions of the said 526 surviving purchasers 
will also be alive at the end of the said 30 years. Now, the com- 
panions of these 520 surviving purchasers were at the time of mak- 
ing the grants just as many as those purchasers themselves — that is, 
526. We must therefore inquire, by M. de Parcieux's table, how 
many of these 526 companions of the said 526 surviving purchasers, 
who were all living and of the age of 20 years at the time of mak- 
ing the grants, will be alive at the end of the said 30 years. Now, 
it appears by M. de Parcieux's table that out of 814 persons at the 
age of 20 years, all living at the same time, only 581 will be living 
at the age of 50 years ; and consequently out of 526 persons of the 
age of 20 years, all living at the same time, only 526 X-ff£, or 375, 
will be alive at the age of 50 years. Therefore, of the 526 com- 
panions of the 526 surviving purchasers, only 375 will be living at 
the end of the said 30 years. Therefore, only 375 out of the said 
526 surviving purchasers will be entitled to receive the said pay- 
ments of one pound each. Therefore, at the end of the said 30 
years, the grantor of the said future payments will have only 375 
sums of one pound each to pay to the surviving purchasers, which 
will be due to those 375 of them whose companions will be then 
alive. And consequently, to the end that the said grantor may be 
neither a gainer nor a loser by the sum total of all his bargains, it 
is necessary that he should receive at the time of making the said 
grants 375 times the present value of one pound, payable at the end 
of 30 years, or, when the interest of money is 3 per cent, or 375 
times the sum which, being continually improved at compound in- 
terest, during the said term of 30 years, at that rate of interest, will, 
at the end of that time, amount to one pound; because, in that case, 
if he improves the said sum of 375 times the present value of one 
pound, so received, at compound interest, at the said rate of 3 per 
cent, during the whole 30 years, it will in that time increase to just 
375 pounds, which is the sum which he will then be obliged to pay 
to the 375 surviving purchasers, who, by the continuance of the 
lives of their respective companions, will be entitled to their several 
payments of one pound apiece. The present value of one pound 
payable at the end of 30 years, when the interest of money is 3 per 
cent, is .411 9 of a pound. Therefore, 375 times £.4119, or £154.4625, 
is the sum which the said grantor of those future payments ought to 
receive, at the time of making the said grants, from all the 774 



154 NOTES ON" LIFE INSUKANCE. 

purchasers of them. Therefore, the sum which he ought then to 
receive from each of the said purchasers is the 774th part of 
£154.4625 — that is, £.1995, or nearly 4 shillings. And consequently 
when he makes only one such grant of one pound, payable at the 
end of 30 years, to a purchaser of 25 years of age, provided a com- 
panion of the purchaser who is of the age of 20 years at the time of 
making the grant shall also be living at the end of the said 30 
years, he ought to receive for it the same sum of T 2 T of a pound, or 
4 shillings. 

Problem: IY. 

To find the sum of money which a purchaser ought to pay for a 
future sum of one pound sterling, to be received at the end of any 
given number of years, if either the said purchaser or another cer- 
tain person (who may be called his companion) shall be then living ; 
the age of the said purchaser and his companion, and the rate of 
interest of money, and the probabilities of the duration of human 
life, being all given. 

A Solution of this Problem in the Case of a Particular Example. 
— Let the rate of interest of money be supposed to be 3 per cent, 
and the probabilities of the duration of human life to be such as 
they are represented to be in M. de Parcieux's table above men- 
tioned ; and let the space of time at the end of which the said sum 
of one pound is to be paid to the purchaser of it, if he is then liv- 
ing, or to his companion, if the purchaser himself is then deceased, 
and his said companion is still alive, be 30 years, and the age of the 
said purchaser 25 years, and that of his said companion 20 years. 
Then, in the first place, we must look into M. de Parcieux's table to see 
how many persons of 25 years of age are there represented to be all 
living at the same time. This number is 774. We must therefore 
suppose that the grantor of the future payments of one pound to 
the purchaser proposed in the question makes at the same time 773 
more such grants of one pound to as many different purchasers, all 
of the same age of 25 years, to be paid to them at the end of 30 
years, or when they shall be 55 years old, if they shall then be liv- 
ing : and if they shall then be dead, but certain other persons (who 
may be called their companions), who are of the same age of 20 
years with the companion of the purchaser mentioned in the ques- 
tion shall be then alive : to be paid to their said companions respec- 
tively — that is, we must suppose that the grantor makes 774 such 
grants in all, including that to the purchaser proposed in the ques- 
tion ; and we must likewise suppose that all these 774 purchasers of 



NOTES ON LIFE INSURANCE. 155 

these future payments of one pound have the same chance, one with 
another, of living any given number of years, or that there is no 
apparent reason for supposing that any one of them is more likely 
to live to any given future age than any other. This done, Ave must 
inquire how many of these 774 purchasers of these remote payments 
will be alive at the end of 30 years, supposing them to die off in 
that interval of time in the proportion mentioned in M. de Parcieux's 
table. Now, it appears by M. de Parcieux's table, that out of 774 
persons of the age of 25 years, all living at the same time, 52 G will 
be alive at the age of 55 years, or at the end of 30 years. There- 
fore, out of said 774 purchasers of these future payments of one pound 
each, 526 will live to the end of the said 30 years ; and, consequently, 
248 will have died in the mean time. But by the conditions of this 
problem (which differ widely from those of the last problem), all 
these 526 surviving purchasers of these future payments will be enti- 
tled to receive them, and likewise all the surviving companions of the 
deceased 248 purchasers. We must therefore inquire, by means of 
M. de Parcieux's table, how many of the companions of the said 
deceased 248 purchasers will be alive at the end of the said 30 years. 
Xow, the number of the companions of the said deceased 248 pur- 
chasers, at the time of making the grant, was 248, each of the said 
purchasers having had one companion, and their age at the time of 
making the grants was 20 years. Now, it appears, from M. de 
Parcieux's table, that out of 814 persons of the age of 20 years, all 
living at the same time, 581 will be living at the age of 50 years, or 
at the end of 30 years. Therefore, out of the 248 companions of 
the deceased purchasers (who were all living at the time of making 
the said grants, and were then 20 years of age), 248 X -ff£, or 177 
will be living at the end of the said 30 years. Therefore, the gran- 
tor, at the end of the said 30 years, will have 526 pounds to pay to 
the 526 surviving purchasers, and 177 pounds to pay to the 177 sur- 
viving companions of the 248 deceased purchasers — that is, he will 
have, in all, 526 and 177 pounds, or 703 pounds, to pay to both. To 
the end, therefore, that the said grantor may be neither a gainer nor 
a loser by the sum total of all his bargains, it is necessary that he 
should receive from all the purchasers of the said future payments, 
at the time of making the grants, 703 times the value of one pound, 
payable at the end of 30 years, when the interest of money is 3 per 
cent — that is, 703 times .4119 of a pound, or £289.5657. Therefore, 
the sum which he ought to receive, at the time of making the said 
grants, from each of the said purchasers, who are 774 in number, is 
the 774th part of £289.5657, or .3741 of a pound. Therefore, the 



156 NOTES OX LIFE INSURANCE. 

sum which he ought to receive from a single purchaser, as the price 
of such a future payment of one p>ound, when he makes only one 
such grant, is likewise £.3741, or Is. 5%d. 

The reasoning used in the solution of the foregoing problem may 
he extended to the valuation of a future payment, to be received a: 
the end of a given number of years in case any one of three per- 
sons, or more, whose ages are given, shall be then alive. In the ease 
of three lives, the additions necessary to be made to the preceding 
solution will be as follows : 

A:\ J- ■.'-.-;-- ' .- " .: '/ \ of the said Value in the Ca.se of a Particular 
Ei::'rnpU. — In the particular example solved, let the right of the 
purchaser, of 25 years of age, to the future payment of one pound. 
at the end of 30 years, be extended to two other persons besides 
himself, instead of one : so that, if either the purchaser himself . 
either of his said companions shall be then alive, the said future 
sum shall be payable by the grantor of it to one of the said three per- 
sons ; and to make the case more clear and definite, let it be sup- 
posed that the older of these two persons is called his first compan- 
ion and the younger his second companion ; and that, if the pur- 
chaser himself is alive at the end of the said 30 years, the said sum 
of one pound shall be payable to him alone, though either or both of 
his said companions should be also living at the same time ; and that, 
if he is then dead, but his companions are both alive, it shall be paid 
to the elder of the two, or his first companion ; and that, if only 
one of them is then alive, it shall be paid to the said only surviving 
companion. And let the age of the said purchasers older or first 
companion be' 20 years, as was supposed in the foregoing example : 
and that of his younger or second companion, 10 years, at the time 
of making the grant. And further, let it be supposed that the 
grantor of the said future payment of one pound makes 7 74 such 
grants of one pound each, to be received at the end of 30 years, to 
as many different purchasers, all of the same age of 25 years as the 
purchaser proposed in the question : and that the grant to the pur- 
chaser proposed in the question is one of those 774 grants: and that 
each of these purchasers has two companions, to wit, an older, 
first, companion of 20 years of age, and a younger, or second, com- 
panion of 10 years of age; and that each of the sums so granted is 
to be paid by the said grantor, at the end of the said space of 30 
years, provided the purchaser himself, or either of his two compan- 
ions, is then alive — namely, to the purchaser himself, if he is then 
alive; or otherwise to the older of his two companions, if they are 



NOTES ON LIFE INSURANCE. 157 

then both alive ; or if only one of them is then alive, to the said 
only surviving companion. 

These things being supposed, it is evident that the number of 
persons that will be entitled to receive these payments of one pound 
each, at the end of the said 30 years, will be greater than in the case 
supposed' in the solution of the foregoing problem, because not 
only all the 526 surviving purchasers themselves, together with' the 
177 surviving first companions of the 248 deceased purchasers, mak- 
ing in all 703 persons, will be entitled to these payments, as they 
were in that solution, but there will also be some surviving second 
companions of the deceased purchasers, who will also have a right 
to them. What the number of these will be, we must now proceed 
to inquire. 

Now, it is evident, in the first place, from the conditions of this 
question, that the surviving second companions of the 526 surviv- 
ing purchasers can have no claim to these payments of one pound ; 
because they are to be made to the said surviving purchasers them- 
selves. And, for the like reason, it is evident, in the second place, 
that the second companions of those deceased purchasers whose 
first companions are alive at the end of the said 30 years can have 
no claim to these payments ; because it is provided that when both 
the companions of a deceased purchaser are alive at the time his 
payment becomes due, it shall be made to the said purchaser's first 
companion, and not to his second companion. Now, it has been 
shown, that out of the 248 purchasers who will have died in the 
course of the said 30 years, there will be 177 whose first compan- 
ions (who were 20 years old at the time of making the grants) will 
be alive at the end of the said 30 years. Therefore, the number of 
the said deceased purchasers whose first companions will also be 
dead before the end of the said 30 years is the excess of 248 above 
177 persons — that is, 71 persons. There are therefore 71 deceased 
purchasers, whose second companions will have a right to receive 
these payments of one pound each, if they live to the end of the 
said 30 years. We must therefore inquire how many of the second 
companions of these 71 deceased purchasers will live to the end 
of the said 30 years, supposing them to die off in the proportion 
set forth in M. de Parcieux's table of the probabilities of the 
duration of human life. 

Now, these companions are evidently 71 in number, because each 
of the said 71 deceased purchasers had one second as well as one 
first companion ; and the age of these second companions, at the 
time of making the grants, is supposed to be 10 years. Now, it ap- 



158 NOTES OK LIFE INSUKAKCE. 

pears from M. de Parcieux's table, that out of 880 persons of the 
age of ten years, all living at the same time, 657 will live to the 
age of 40 years, or to the end of the term of 30 years. Therefore, 
out of the said 71 second companions of the said deceased pur- 
chasers, there will be 71 X fw, or 53 > wno wil1 live to the end of 
the said 30 years. Therefore, the whole number of persons enti- 
tled to receive the said payments of one pound each, at the end of 
the said 30 years, will be, first, the 526 surviving purchasers, and, 
secondly, the 177 surviving first companions of the 248 deceased 
purchasers, and, thirdly, the said 53 surviving second companions 
of those 71 of the said 248 deceased purchasers, whose first com- 
panions will have died before the end of the said 30 years — that is, 
in all, 756 persons. Therefore, at the end of the said 30 years, the 
aforesaid grantor will be obliged to pay to the said surviving pur- 
chasers, and to the said first and second companions of the pur- 
chasers that are deceased, the sum of 756 pounds. Therefore, to the 
end that, when the said payments become due, the said grantor 
may be neither a gainer nor a loser by the sum total of all his bar- 
gains, it is necessary that he should receive, at the time of making 
said grants, 756 times the present value of one pound certain, pay- 
able at the end of 30 years, when the interest of money is three per 
cent — that is, 756 times .4119 of a pound, or £311.3964, from all the 
774 purchasers of these future payments. Therefore, the sum which 
each of the said purchasers ought then to pay him, as the price of 
the said future payment of one pound to be received at the end of 
the said 30 years, is the 774th part of £311.3964, or .4023, or 8s. \d. 
And, consequently, this sum of 85. \d. is likewise the price which 
a single purchaser ought to pay for a grant of such a future pay- 
ment of one pound, to be received at the end of 30 years, if either 
himself or either of his two companions aforesaid shall be then 
alive, when the grantor of it makes only one such grant. 



METHOD OF COMPUTING A^^UITIES. 

" A very easy and convenient method of deducing the value of a 
life annuity of one pound a year for a life of any given age from 
the value of the same annuity for a life that is older than the for- 
mer by one year : by the help of which method, a whole table of 
the values of a life annuity of one pound a year for every age of 
human life, proceeding from the older ages to the younger by the 
constant difference of a year, may be computed with nearly the 
same labor as is necessary to obtain the value of the same annuity 



NOTES ON LIFE INSURANCE. 159 

for the first, or youngest, life in the table. This method was first 
eommunicated to me by Dr. Price ; but it was published in the 
year 1779 by Mr. William Morgan, the actuary to the Society for 
Equitable Assurances near Blackfriars Bridge, in his treatise on the 
Doctrine of Annuities and Assurances on Lives, pages 56, 57 ; and it 
had been published before by Dr. Price himself in his Treatise on 
Reversionary Payments, ITote O of the Appendix, and likewise by 
Mr. Thomas Simpson, in his book of Life Annuities, Prob. I., Coroll. 
7 ; which last book was published so long ago as the year 1742. 
But I should suspect that it was not known to Mr. De Moivre, 
when he calculated his tables of the values of life annuities ; for, 
if it had, I should imagine he would hardly have thought it neces- 
sary to have recourse to a certain inaccurate hypothesis concerning 
the probabilities of life, in order to diminish the labor of his com- 
putations, which would have been almost equally facilitated by the 
use of this excellent method." — Masseres, 1783. 



TETENS, OF KIEL. 

In 1785, Prof. Tetens, of the University of Kiel, published a 
method of computing commutation columns for use in making life 
insurance calculations. In this he introduced the device of multi- 
plying both the numerator and denominator of the fraction which 
at each age represents the amount that will effect insurance for 
life, by v raised to a power equal to the ages of the insured ; v 

being equal to , and r being the rate of interest. Tetens com- 
menced the computations at the oldest age in the table of mortality, 
and in regular order took an age one year younger to the youngest 
age given in the table. This method was first used in calculating 
the value of a life series of annual payments ; but the principles 
apply equally to insurance. 

The method of Tetens makes these calculations simple and easy. 
But if the real meaning of the commutation columns is not under- 
stood, they are of no use whatever to an ordinary business man. 
Masseres was not acquainted with this method, it being introduced 
after he wrote ; but the clearness with which he explains, and his 
evident determination to make himself understood by his readers, 
form a marked contrast with the writings and publications of 
some actuaries in these days. 



160 >-OTE5 OX LIFE INSURANCE. 



QUOTATION — 1870. , 

A life insurance actuary says, in a work published in 1870 : "The 
object of this treatise is to explain the science of life insurance, so 
that its main features can be easilv understood bv anv one havinc 
an ordinary knowledge of arithmetic." In explaining the commu- 
tation columns, he says. ;i In this treatise, D represents the number 
of the living at any age, discounted, at four per cent, for the num- 
ber of years corresponding to that age. For the sake of conve- 
nience and uniformity, we employ that power of T .^ in computing 
the D column which corresponds to the given ages ; X represents 
the sums of the discounted numbers of the living from ninety-nine 
up to any given age ; C denotes the number of the dying, discount- 
ed by the same power of T \^ increased by one, as in the D column ; 
and M, computed like the X column, is the discounted numbers of 
the dying at the various ages. In computing the C column, we 
use a power of T ^ greater by one than the corresponding age, 
since the losses by death are regarded as taking place at the end 
of the vear. The numbers in these columns have the same relative 
value to each other, and the same results are produced as if we 
took the first power of T ^ for the age ten, or any required age, 
and so on.*' 

Reflect upon this. " Discount the number living." Remember that 
the avowed object of this actuary was "to explain the science of life 
insurance," so that it can be understood by any one having an or- 
dinary knowledge of arithmetic. Read again his explanation of 
the commutation columns ; and then consider the following state- 
ment, published in 1553, by a distinguished actuary : ,; It is not to 
be expected that men who enjoy honor and emolument from being 
considered the exclusive depositaries of a science so useful to the 
world, should so popularize and simplify it as to remove the bread 
from their own mouths and the glory from their own wigs."' 

The same distinguished actuary, in 1871, says in substance : 

Having curtate commutation columns in which a?+w is the age 
at which a policy becomes certainly payable, a constant annual 
premium being payable till that age, and writing x+n as the " argu- 
ment of the summation" when we assume /,_ a =0. The expression 

for the net single premium at age x in this case is 1 — (1 — l ') "-k ' ■> 
and that for the net annual premium is — ^ (1 — v). 



NOTES OK LIFE INSURANCE. 161 

The reserve at the end of t years is expressed by a+n II, +{ . This is 
equal to the net single premium at age x + t, less the value at that 
age of the net annual premiums yet to be paid. Hence, 



, + .H, +1=1 -( 1 _^-(3 : -( 1 -,))x^ 



+* 



From which : 

We now quote his exact words, 1871 : 



D ± ± „N, + « 



" SELF-INSURANCE. 

H Since the normal reserve which will exist on a policy at the end 
of any year is just so much more in the hands of the company 
towards the payment of the claim that might have occurred in that 
year than if the insured had each year paid only its risk, it may 
properly be called a self-insurance. Hence H x+t may be called the 
self-insurance value of the policy. It may be regarded as a savings- 
bank deposit, with only this difference, that it can not be withdrawn 
till the expiration of the policy by death, or what, in case of an 
endowment policy, we have assumed to be death. If x + n is the 
limiting age up to which some may live, but none beyond, by the 
assumption we should have H^=l. And if S is the sum insured, 
the self -insurance will begin with SH X+1 the first year, and increase 
from year to year until it is S in the last year. Consequently the 
insurance done by the company per dollar of the policy is a series 

of complementary risks, 1 — H x+1 , 1 — H x+2 , 1 — H x+3 , O. 

The normal cost or contribution to claims of each of these risks, at 
the end of the year in which it takes effect, is, 

f(l-H I+1 ), f±i(l-H I+! ), etc. 

^x C x+l 

u This is what becomes of the net premiums and assumed interest 
thereon, so far as they do not go to form the self-insurance deposits 
aforesaid ; that is, H x+1 , H x+2 > etc. Hence the present value of all 
the normal contributions, which a policy is liable to make for the 
settlement of claims other than its own, may be called the 

Insurance Value. 

" To conduct a company either with reference to permanent profit 
as a stock company, or equity and stability as a mutual one, with the 
present great variety of policies, it is almost as important to know 



162 NOTES ON LIFE INSURANCE. 

this value as the self -insurance or reserve. The new tables, which 
will assist in calculating this value for all single-life policies, are 
constructed on the following principles, and their use will be pre- 
sently illustrated : 

" If x -f n be the age at which a policy becomes certainly payable, 
a constant annual premium being payable till that age, referring 
to the D and N columns (see tables), we shall have the successive 
self -insurance values in terms of these tables, thus : 

1st. ,+.H^,=l \ x -^±i. 

2d TT — i ^* v t+^'+s 
^u. ^»£L^2 — 1 ^-r X — -p: 9 

etc., etc. 

Consequently we have the insurance done by the company each 
year: 



1st. 1— H^., = -^x 



x+n-^x ■ u x+l 

"D "NT 

^ Q ' -l z+n&z+'i — xf~ X — pj > 

z-j-n^ * Uz+2 

etc., etc. 

The values of these risks will be, 

1st. f (i-^H \=^i- x ^«x* 



2d. ^X-^Hj^x^x^i, 

etc., etc. 

" The first of these is certain because the premium is paid in ad- 
vance, but must be discounted by the factor v to refer it to the be- 
ginning of the year. The second must not only be discounted two 

years, but must be multiplied by the fraction -y^, expressing the 

probability of the party being alive to pay the second premium. 

Hence, the discount factor will be v 2 -y^. In like manner, the dis- 

'x 

count factor for the third year will be v* -—^, and soon. Observing 

'x 

that — ^=- is a factor common to every term of the series to be dis- 

counted, and substituting for D^, D^ 2 , etc., their values, v z+I 4 +l3 
v* +s 4+*, etc., and applying the discount factors above explained, we 
have the insurance value, which we will designate by the symbol I. 



NOTES ON LIFE INSURANCE. 103 

' + " 1 '-^n:W+% +1 x i a + 5^U X l. +1 x4 + etc 'j 

" Multiplying numerators and denominators by v% 

T _ P* / uXt., v «r^* ^_ x+ JST, + t v x+ > d t+l xl x+l , _ \ 

Canceling like factors in numerator and denominator, and substi- 
tuting D, for 0*4, we have, 



-^* ,.*+»■" a+1 /n / Ti+ii^,+!A , 



Dx v ,+.N, +1 x r jl +, + „N I+2 x 7 ^ ±I + etc. 
N" X 2±1 *±! (7) 

"Obviously, if we assume #=10 and perform all the multiplica- 
tions of -~ into N n , y 1 - into N" ia , etc., as indicated in the nume- 

rator of the last factor of (7), the summation of the products (after 
the manner in which the N column is produced from the D) will 
produce a column of numerators which we will call A (lucus a ?ion) y 
and this will give us, instead of (V), 

T — D * y * +nZ ** nnd (P,\ T — P * y *±gjf± l 
a+n** — XT A 1^ » V / *+ nL *+* — XT * T> » 

a+n-^a *-> x a+n-^ a ■ lJ x+t 

when the policy has completed t years and the (t+l) th premium is 
just paid. 

" Similarly, to find the insurance value of a paid-up policy, express- 
ing the self-insurance values in terms of the D and N" columns, we 
shall find the aggregate of the discounted future costs of insurance 

reduce itself to x +J x+t = (l—v) x+ " x+t . 

v x+t 

" To get the insurance values of limited-premium life or en- 
dowment policies, let 0, be the premium to be paid q times, and 

v — a =c„ c x — H z+ ,= c 2 , c, — H z+i = c 3 c x — B x+q _ x =c r 

Then referring to Table XXIL, p. 200, column d x , and denoting the 
insurance value when the first premium is just paid by a+n I»i„ we 

shall have , + „I, l ,= - ' Cl + <? - +lC " T :- " + J ^-- c v + g^ X '-#±i. Af . 

Mb Me Ux+q 

ter having thus found the initial insurance value, we may find that 
of the succeeding years by applying the coefficient of accumulation, 
thus : 



*+n x *+2 | q 

etc., etc., etc. 



a+n A a+l| ? — ; \ x+n*-x | q j <-\ j 

*x+l \ t'x+l I 

T —J&±{ T , ^+V \ 

. A *+2| 7 T i a+n^x+ll? 7 C 2/> 

*a+2\ *a+2 / 



164: NOTES ON LIFE INSURANCE. 

" For the following briefer process applicable to any policy payable 
at the age x + n or previous death, including, of course, the ordinary 
whole-life policy, the public are indebted to the keen analysis of 
-y[ r * * * * jjjg notation is slightly modified to adapt it to the 
tables. 

" Let <f> 9 be the annual premium to be paid q times, and when the 
first premium is just paid, we have : 

and when the (£-|-l) th premium has just been paid, 

T j x+n^x+t ! A x+q^x+t 

x+nK+t | v — a ~J\ r YVy; 

■ lJ z+t Ux+t 

Though the insurance value of an annual premium term policy is 
greater than that of an endowment policy for the same term and 
amount, there is, except for a very long term, too little reserve on 
it to secure a proper surrender charge. Therefore, unless the pre- 
mium is extravagantly loaded, the company is not sufficiently se- 
cured against loss by lapse after having paid the usual commission. 
For this reason, and because this sort of policy is little sought by 
the public which fails to see value, except in the indemnity or en- 
dowment, it is hardly necessary to insert any formulas for the insu- 
rance values of term policies. But as these policies are really valu- 
able, and when paid for by single or a limited number of annual 
premiums, the reserves afford sufficient security to the company 
paying a moderate commission, it may be well to provide for busi- 
ness which may arise. Putting K for the insurance value of policies 
of this class, we have for the ordinary term policy of n years when 
the (£ + l) th premium is just paid, 

XT _*+n +l4+ t / y+nA+t / g % 

" If the premium is limited to q payments, in which case it is 



x+n7x ■%£ — Px) 

X+q^X 



the value is, 



TT x+n+\A x+t n% x + n A x+ t x+qA*+t 

x+n*±x+t\q — — j} V j. -T^-rj ' 



'x+t -"x+t 

and a single premium term policy is, 



-T7- x+n+\A x+t x +n"x+t 

x+n**-x+t— fZ V j\ 

l-'x+t XJ x 



c+t -"x+t 

"The insurance value of a pure endowment or tontine policy is, 



NOTES ON LIFE INSURANCE. ,165 

of course, negative, the operation being wholly the exact reverse of 
insurance, to wit, of term insurance. For example : if T be the 
insurance value of a tontine, or pure endowment policy, at annual 

premium, we have, by substituting in (8) for — ^_, its value =d+ 

xf „Tr c =;7+ x+ »y, + * + ,A, and subtracting (9) 

T — (\ _L /> \ x + n" x+t x+n+l^ x+t 

x+n X x+t — \L -tx+n^x) yx Tv J 

which is always negative, and if paid up it becomes, 

rp xj-n^x+t x+n+\^x+t 

x+n±x+t— tv 1) * 

" It must not be inferred from the negativeness of the insurance 
value of endowment that the company loses or is weakened by it. 
The mortality being normal, it neither gains nor loses. It gains if 
the mortality is greater, and loses if it is less. But the insurance of 
pure endowment being inverted, its negativeness indicates that the 
interest of the policy-holder as a probable survivor is to have the 
vitality of the company diminished. He can afford to pay less than 
nothing to increase it. Hence, a fortiori, he can not be satisfied if 
he is charged for expenses more than his x+n H. x+t would cost him in 
a savings-bank, unless there should have occurred in the company 
that extraordinary mortality which is the only source of prosperity 
to pure tontine companies. This is not to be hoped or wished for 
when endowment is coupled with insurance in the same policy." 



166 NOTES ON LIFE INSURANCE. 



CHAPTER XIV. 
ALGEBRAIC SUMMARY AND FORMULAS. 

7 X = tabular number living at any age, x. 

l z+n = tabular number living n years after the policy was issued at 
age x. 

d x = tabular number of deaths between age x and age jc + 1. 

^*+*=tabular number of deaths between age x+n and age 
x + n + 1. 

r=rate of interest per annum. 

0= =the sum that will, if invested at the rate r, amount to 

$1 in one year. 

v x =[- 1 =the sum that will, if invested at the rate r. com- 

pounded annually, amount to $1 in x years. 

XET PREMIUMS. 

^— *=the amount that will at age x insure $1 for one year. 

v- ' + v*-yl=the amount that will at agese insure |1 for two years. 

v-j*+v s -y^ + other similar terms to table limit = the amount that 

will at age x insure $1 for life. Continuing these yearly terms to 

age 95 (American Experience oldest table age), and introducing 

the factor v x in both numerator and denominator, we have, 

v x + 1 d x + v x+ >d x+l + + v 96 d« 5 . 

— = -=the amount that will at age x 

0*4 

insure $1 for whole life. Designating the terms of the numerator 

by Q,., C x+1 C 95 , and the denominator by L\, we have, 

C +C + c 

— * +1 " " ' — -=the amount that will at age x insure $1 

for whole life. Call the sum of all the terms of the numerator M a 

M 

and the expression becomes =r* The net single premium that will 

M j. 

at age x + n insure $1 for whole life =-=^-?. 



NOTES ON LIFE INSURANCE. 107 

The amount that will at age x insure $1, the insurance to com- 

v*d x+1 v x+ *d x+1 



raence at age x + 1, and continue for one year, is — f— 

Q 

-— -. The amount that will at age x insure $1, the insurance to 
commence at age x + 2, and continue for one year, is — ^±1— _f±? 



Qr+2 



= ~-. The amount that will at age x insure $1, the insurance to 
commence at age cc + 1 and continue for whole life, is expressed by 

In like manner, the amount 



C I+1 + C xf2 + +C 95 M, +1 



that will at age x insure $1, to commence at age sc + 2, and continue 

forwholelif^is ^ 4 " ^ 4 " ' + C 95 = ^ Therefore the 

amount that will at age x insure $1, to commence at age x ; 
82, to commence at age ce-j-1 ; and so on, increasing the 
amount insured for whole life $1 each year, is expressed by 



M, + M x+t + +M 95 



Call the sum of all the terms of the 



numerator R,, and we have -pr x =the amount that will at age x in- 
sure $1 the first year, $2 the second, $3 the third, and so on each 
year, increasing the amount insured $1 each year to the table limit 
of age. 

The amount that will at age x insure $1 for life, the insurance to 

begin at the end of n years, is ~~ ; therefore the amount that will 

at age x insure $1 for n years only is - x ' x+n . 

The amount that will at age x insure $1 at age x + n, $2 at age 

x + n + 1, and so on, increasing the amount insured $1 each year to 

■p 
the table limit, is -y^- The amount that will at age x insure $1 

the first year, $2 the second, and so on, increasing the amount in- 
sured $1 each year for n years, after which the insurance is to 
cease increasing and remain equal to n times $1 to the table limit, 

is = y? 2 . The amount that will at age x insure n times $1, 

the insurance to commence at age x + n and continue to the table 

limit, is ^-*-±-». Therefore ~* — ^ — ^^t? = the amount that 



168 NOTES OS LIFE INSUKANCE. 

will insure $ I the first year, $2 the second, and so on, increasing the 
insurance $1 each year for n years, the insurance to cease at that 
time. 

Designate by Z* the net single premium that will at age x insure 

M, . 

$1+ this premium. Then since ~ is the amount that will insure 

M 

$1 ; (1 +Z X ) ~ is the amount that will insure 1 -f Z a . Hence Z* = 

[1+Z, )~ x ; from which Z x =p^M • 

The net single premium that will provide the insurance of $1 for 
life, and return this net premium plus m per cent thereof, is ex- 

pressed by j^-^-. 

The net single premium that will at age x secure the payment of 
$1 at age x + z, in case the insured is then alive, is expressed by 

v%^ v*+%+ z D I+ , 



I. v% T> x 



age x + z, or at death if prior 



net single premium at age x to secure $1 at 
Dr. 
=net single premium at age x to secure 



$1 at age x + z, or at death if prior, and return this premium plus 
m per cent of itself. 

Value at age x of a Life Series of Annual Payments of %\ each, 
the first in advance. — The value at age x of $1, to be paid in one 
year, provided the person is alive to make the payment, is expressed 

by r-y^. The value at age x of $1, to be paid in two years, provid- 

ed the person is then alive to make the payment, is r 5 -^. And in 

like manner for each succeeding year to the table limit. Adding 
together all these yearly values, noting the condition that the first 
payment of $1 is to be made immediately, and we have, 

l x J rvl^ l + v'l x +«-\- ... to table limit 



This is the amount that will if paid at age x be the equivalent of 
a life series of annual payments of $1 each, the first in advance. 



NOTES ON LIFE INSURANCE. 169 

Multiplying the numerator and denominator by v*, the expression 
becomes, 

v x l z + v' +l l x+ i + v' + 'l, + , .... +tt 9 V 95 

v% 

Designating the numerator of this fraction by N„ and noting that 

the denominator has already been designated by D x , and we have 

N 

— -*=the value at age x of a life series of annual payments of $1 

each, the first being made in advance. 

The value at age x of a life series of annual payments of $1 
each, the first payment to be made at age se + 1, is expressed by, 

V* +l l x+ ,+V* + % + i + +V'%S = X X+1 

v% - D, ' 

In like manner, the value at age xof a life series of annual pay- 

N 
ments of 81 each, the first to be made at age <c + 2, is -yr^, and so 

on for each succeeding year to the table limit. Adding together all 
these respective values, and designating the sum of the terms of the 
numerator by S x , we have, 

S J _ N J + N" x+1 + N :e+2 + .... +K, 5 _ 
D." D, - 

The value at age a; of a life series of annual payments of $1 im- 
mediate, $2 at the beginning of the second year, $3 the third, etc., 
increasing the payments $1 each year to table limit. 

NET ANNUAL PREMIUM. 

The amount that will at age x insure $1 for whole life has been 

M 

found to be equal to =r-. The amount in hand that is the equiva- 
lent of a life series of annual payments of $1 each is equal to 

N 

— ". Designate the net annual premium that is the equivalent of 

M 

the net single premium — ' by aP xi and we have the proportion 

1ST M M 

ff ' fr • • $* : a -P*> or aPx=-^ =net annual premium to secure $1 

at death or table limit. 

The value at age x of a series of annual payments of $1 each 

ET N z 

for a years is expressed by * x+a . From the following pro- 

L'a 



170 NOTES ON LIFE INSURANCE. 

portion, we find ^'^^ ; ^; ; $1 : M ' . Therefore x . M ' 
1 V x T> x N t —N x+a N,— N.^ 

is the net annual premium for « years to secure $1 at death, or at 

table limit of age. 

]\1 -\f _ 

— A — ^-^net annual premium to insure $1 for z years, the in- 

-^ X -*^ x+z 

surance to cease at the end of that time. 

Designate by Z' x the net annual premium to secure $1 at death 
and a return of all the net annual premiums paid. Then, since 

^Y is the amount that will insure $1 the first year, $2 the second, 

7' R 

and so on, -~- * will insure 7*' x the first year, 2Z' S the second, and so 

on, increasing the insurance by Z' x each year to table limit ; there- 

fore v, + ^T- z * x D,' oiZ -=W=r; 

The net annual premium that will insure $1 at death, and return 
all the net annual premiums paid plus m per cent thereof, is 

n— (i+»i)r; 

=r= — ^= j- — , /T -, ^ — r = net annual premium for a years 

N.—N^ — (1 + m) (R — R, +a ) l 

to secure $1 at death, and the return of all these net premiums 

plus m per cent thereof. 

^ — ^±f— —net annual premium to secure $1 at age x+z if the 

assured is then alive. 

== — ^— =net annual premium for a years to secure $1 at age 

^x -^ x+a 

x + z if the insured is then alive. 

— — = - ^FFi — Ti tf — ^ = net annual preinium at 

N,— N^— (1 + m) (R — R* + — zM x+z ) * 

age x, to secure $1 at age x+z f if the assured is then alive, and to 

return the loaded premiums in case of death prior to age x+z. 

U x — M^ + D*. 



81 at age x-\-z, or at death, if prior. 



net annual premium for a years, to secure 

net annual pre- 



z+z 



X — X x+Z - (l + i?i) (R 2 — R i+ — zK x ^+zD x+s ) 
mium to secure $1 at age x+z, or at death, if prior, and in either 
event to return these premiums plus m per cent thereof. 



NOTES ON LIFE INSURANCE. 171 



FORMULAS FOR THE DEPOSIT OR "RESERVE." 

The net value of a policy at the end of any policy year may be 
found, in case the net value at the end of the preceding year is 
known, by either of the three following general formulas : 

1. Hx+» = Hh*-i ( h -+«-i + « p x — £,-,-„-,). (Sec page 96.) 

2. H x+ft = a x+n _ x (11,+^t 4- aP x ) — k x+n _ x . (See page 97.) 

3. H.+., = 1 — u x+n _ x (v — a? x — II, + „_,). (See page 98.) 

In case the deposit at the end of the preceding year is not known, 
it may be calculated directly in different cases by the following for- 
mulas : 

M 

4. =-^ = net value at the end of n years of a full-paid whole- 

life policy. 

5 « t^ m — f -y L — - — rrr^r = net value at the end of n years of a 

T> x+n [D— (l+w)MJ J 

full-paid whole-life policy, to secure $1 and the return of the pre- 
miums paid ; m being the percentage, or loading, added to the net 
premiums. 

]yj jyj 

6. J+ " — = net value at the end of n years, term policy 

for z years, full paid. 

7. j^- = reserve at the end of n years, to secure $1 at age x+z, 

if the insured is then alive, full paid. 

D a+r D— (14- m)M, + (l+m)Mn _ w = ^ 



D — (1 + m)(M—M x+! ) ~ D x+n 

serve at the end of n years, full paid, to secure $1 at age x+ z, if the 
insured is then alive, and in case of prior death, the premium actually 
paid to be returned at the end of the year in which the insured dies. 

9. r+n fr^ — = deposit at the end of n years, full paid, 

to secure $1 at age x-\-n, or at previous death. 

10. =- — X t^ 7z mi — tf — ^r — ^ = reserve at end ot n 

D x+n D,— (1 + m) (M — M x+Z + B x+Z ) 

years, full paid, to secure $1 at age x-\-z, or previous death, in either 
event the premium actually paid to be returned. 



172 NOTES ON LIFE INSURANCE. 

11. =p=^- — ^ x^y^=l — ^-^ = reserve at ageas+w, life po- 

licy, annual premiums. 

12. =- — ^ — X ^ = reserve at a^e x-tn. lite 

policy, annual premiums for a years, and before a years have elapsed. 
After a years, the formula for full-paid applies. 



- 1 «• ^ ^r t^ x -r-> - -, ^r > = re- 



d^» *r«— (i+»»)R, '. p,_, j ~~ 

serve at end of n years, annual premiums, to secure $1 at death, and 
return all the premiums paid. 



IW. X - X,_ .— (1 4- m) (R,— R_) X 



; y._ — y .. — (i - m ) < r.._.— R^ a + hM.._„ 



1= 



reserve at end of n years, and before a years have elapsed, annual 
premiums for a years, to secure $1 at death, and return all the premi- 
ums actually paid. 

15. =r — < 1 -f ^t ~ ! — - — - — r-r^ ^ — r .- = reserve tor 

D,- » ( ^ , — > ;-* — (1 - m) (R, — R r _.J ) 

above policy when a or more years have elapsed. 

16. ' * — — -^ ^ X — ^ — = reserve at end 

of // years, annual premiums for z years, insurance to cease at that 
time. 

17. 5^ — ^TL X X "~* ~ y '"' = reserve at end of n 

"*-^z— » -*-* z —^ z—z U z—n 

years, annual premiums, to secure $1 at age x— 2, if the insured is 
then alive. 

18. =-^ — ^ ^ — X ^ — = reserve at end of n years, 

before a years have elapsed, annual premiums for a years to secure 
%\ as above. 

D. . X — X._ n — (1^//?)(R.— R,_ s — «M,_) 

. 19. =±zl X * ' \ — r-ET — ^ ^ = reserve at end 

D M ^ — N r _— (1 -//<) (R s — R_ s — zM,_ s ) 

of n years, annual premiums to secure $1 at age sb+ 2, if the insured 
is then alive, the premiums paid to be returned if death occurs prior 
to a^e sc+a 



NOTES ON LIFE INSURANCE. 173 

90 M «+» ~ M ' + ' + P - + ' M * ~ M *+' + P '+' v ^ ~ N ' + ' 
D, +n N, - N, + , D, +u 

reserve at end of w years, annual premiums for z years, to secure $1 

at age x + z, or at death, if prior. 

M, + . — M f+ , + D J+I _ M f - M, + . + D, f . N x+H — N x+a _ 



reserve at end of n years, before a years have elapsed, annual pre- 
miums for a years to secure $1 at age x + z, or at death, if prior. 
After a years, it becomes full paid. 



ANNUITIES. 

1ST 

22. ~- = net single premium at age x to secure $1 annually for 

life. Payments to be made at beginning of each year ; the first im- 
mediate. 

N 

23. =~^ = reserve on above at the end of n years, just before 

the n-\-\ payment of the annuity is made. 
N" n 

24. — — = net single premium to secure an annuity of $1 

for z years. 

25. J+n — = reserve for above at end of n years. 

26. -^y 1 = net single premium at age x, to secure an annuity of 

$1 ; first payment to be made at the end of z years, and annually 
thereafter. 

Note. — The formulas in use for calculating net premiums and 
net values for joint life and survivorship policies are quite similar 
to those for single lives. The commutation columns for joint lives 
are very voluminous. They are given, together with formulas used 
in their application, in a work entitled " Commutation Tables," 
published by C. & E. Layton, 150 Fleet street, London, England, 
1858;* and in " Commutation Tables," published by S. W. Green, 
16 Jacob street, New-York, 1873 ; also in other life tables. 



NOTES ON LIFE INSURANCE. 175 



LIST OF TABLES. 



I. Amount of $1 at the end of x years, at 4 and 4^ per cent. 
II. Present value of $1 due in x years = v x , at 4 and 4| per 
cent, with corresponding logarithms. 

III. Actuaries' Table of Mortality (with percentage of deaths). 

IV. American Experience Table of Mortality (with percentage 
of deaths). 

V. D, N, S, C, M, R columns, Actuaries' 4 per cent. 
VI. D, N, S, C, M, R columns, American Ex., 4^ per cent. 
VII. A x , Actuaries' 4 per cent, and American Ex., 4-J per cent. 
VIII. u xi c x , k xi Actuaries' 4 per cent. 
IX. ie xi c xi k x , Am. Ex. 4^ per cent. 
X. Decimals of a year. 



176 



NOTES ON LIFE INSUKANCE. 



Amount of $1 in any Number of Years, 





4 per cent. 


4% per cent. 


1 


4 per cent. 


4>£ per cent. 


1 


1.040 0000 


1.045 0000 


51 


7.390 9507 


9.439 1049 


2 


1.081 6000 


1.092 0250 


52 


7.686 5887 


9.863 8646 


3 


1.124 8640 


1.141 1661 


53 


7.994 0523 


10.307 7385 


4 


1.169 8586 


1.192 5186 


54 


8.313 8143 


10.771 5868 


5 


1.216 6529 


1.246 1819 


55 


8.646 3669 


11.256 3082 


6 


1.265 3190 


1.302 2601 


56 


8.992 2216 


11.762 8420 


7 


1.315 9318 


1.360 8618 


57 


9.351 9105 


12.292 1699 


8 


1.368 5690 


1.422 1006 


58 


9.725 9869 


12.845 3176 


9 


1.423 3118 


1.486 0951 


59 


10.115 0264 


13.423 3569 


10 


1.480 2443 


1.552 9694 


60 


10.519 6274 


14.027 4079 


11 


1.539 4541 


1.622 8530 


61 


10.940 4125 


14.658 6413 


12 


1.601 0322 


1.695 8814 


62 


11.378 0290 


15.318 2801 


13 


1.665 0735 


1.772 1961 


63 


11.833 1502 


16.007 6027 


14 


1.731 6764 


1.851 9449 


64 


12.306 4762 


16.727 9449 


15 


1.800 9435 


1.935 2824 


65 


12.798 7352 


17.480 7024 


16 


1.872 9812 


2.022 3701 


60 


13.310 6846 


18.267 3340 


17 


1.947 9005 


2.113 3768 


67 


13.843 1120 


19.089 3640 


18 


2.025 8165 


2.208 4788 


68 


14.396 8365 


19.948 3854 


19 


2.106 849-2 


2.307 8603 


69 


14.972 7099 


20.846 0628 


20 


2.191 1231 


2.411 7140 


70 


15.571 6183 


21.784 1356 


21 


2.278 7681 


2.520 2412 


71 


16 194 4831 


22.764 4217 


22 


2.369 9188 


2.633 6520 


72 


16.842 2624 


23.788 8207 


23 


2.464 7155 


2.752 1663 


73 


17.515 9529 


24.859 3176 


24 


2.563 3042 


2.876 0138 


74 


18.216 5910 


25.977 9869 


25 


2.665 8363 


3.005 4345 


75 


18.945 2547 


557.146 9963 


26 


2.772 4698 


3.140 6790 


76 


19.703 0648 


28.368 6111 


27 


2.883 3686 


3.282 0096 


77 


20.491 1874 


29.645 1986 


28 


2.998 7033 


3.429 7000 


78 


21.310 8349 


30.979 2326 


29 


3.118 6514 


3.584 0365 


79 


22.163 2683 


32.373 2980 


30 


3.243 3975 


3.745 3181 


80 


23.049 7991 


33.830 0964 


31 


3.373 1334 


3.913 8574 


81 


23.971 7910 


35.352 4508 


32 


3.508 0588 


4.089 9810 


82 


24.930 6627 


36.943 3111 


33 


3.648 3811 


4.274 0302 


83 


25.927 8892 


38.605 7601 


34 


3.794 3163 


4.466 3615 


84 


26.965 0047 


40.343 0193 


35 


3.946 0590 


4.667 3478 


85 


28.043 6049 


42.158 4551 


36 


4.103 9325 


4.877 3785 


86 


29.165 3491 


44.055 5856 


37 


4.268 0899 


5.096 8605 


87 


30.331 9631 


46.038 0870 


38 


4.438 8134 


5.326 2192 


88 


31.545 2416 


48.109 8009 


39 


4.616 3660 


5.565 8991 


89 


32.807 0513 


50.274 7419 


40 


4.801 0206 


5 816 3645 


90 


34.119 3333 


52.537 1053 


41 


4.993 0614 


6.078 1009 


91 


35.484 1067 


54.901 2750 


42 


5.192 7839 


6.351 6155 


92 


36.903 4709 


57.371 8324 


43 


5.400 4953 


6.637 4382 


93 


3S.379 6098 


59.953 5649 


44 


5.616 5151 


6.936 1229 


94 


39.914 7942 


62.651 4753 


45 


5.841 1757 


7.248 2481 


95 


41.511 3859 


65.470 7917 


46 


6.074 8227 


7.574 419> 


96 


43.171 8414 


68.416 9773 


47 


6.317 8156 


7 915 2685 | 


97 


44.898 7150 


71.495 7413 


48 


6.570 5282 


8.271 4556 


98 


46.694 6636 


74.713 0496 


49 


6.833 3494 


8.643 6711 


99 


48.562 4502 


78.075 1369 


50 


7.106 6833 


9.032 6363 


100 


50.504 9482 


81.588 5180 



NOTES ON LIFE INSUKANCE. 



177 



Present Yalue of a Dollar due in any Number of Years and 
eorrespon ding Logarithm. 





v 4i t. 


U x 4£ %. 


B»4*. 


Iv* 4 %. 




V* 4£ *. 


hv* 4£ *. 


f 4*. 


ho? 4 *. 


1 


.9569378 


1.9808837 


.9615385 


1.9829667 


51 


.1059422 


.0250692 


.1353006 


.1312997 


g 


.9157300 


.9617674 


.9245562 


.96593:53 


52 


.1013801 


-.0059529 


.1300967 


.1142664 


3 


.8762966 


.9426511 


.8889964 


.9489000 


53 


.0970145 


2.9868366 


.1250930 


.0972J330 


4 


.8885618 


.9235348 


.8548042 


.9318666 


54 


.0928368 


.9677203 


.1202817 


.0801997 


5 


.8024510 


.9044185 


.8219271 


.9148333 


55 


.0888391 


.9486040 


.1156555 


.0631663 


6 


.7678957 


.8853022 


.7903145 


.8978000 


56 


.0850135 


.9294877 


.1112072 


.0461330 


7 


.7848285 


.8661859 


.7599178 


.8807666 


57 


.0813526 


.9103714 


.1069300 


.0290997 


8 


.7031851 


.8470696 


.7306902 


.8637333 


58 


.0778494 


.8912552 


.1028173 


_. 01 20663 


9 


.6729044 


.8279533 


.7025867 


.8466999 


59 


.0744970 


.8721389 


.098S628 


2.9950330 


10 


.6439277 


.8088371 


.6755642 


.8296666 


60 


.0712890 


.8530226 


.0950604 


.9779996 


11 


.6161987 


.7897208 


.6495809 


.8126333 


61 


.0682191 


.8339063 


.0914042 


.9609663 


10 


.5896638 


.7706045 


.6245970 


.7955999 


62 


.0652815 


.8147900 


.0878887 


.9439330 


13 


.5642716 


.7514882 


.6005741 


.7785666 


63 


.0624703 


.7956737 


.0845083 


.9268996 


14 


.5399729 


.7323719 


.5774751 


.7615332 


64 


.0597802 


.7765574 


.0812580 


.9098663 


IS 


.5167204 


.7132556 


.5552645 


.7444999 


65 


.0572059 


.7574411 


.0781327 


.8928329 


16 


.4944693 


.6941394 


.5339082 


.7274666 


66 


.0547425 


.7383248 


.0751276 


.8757996 


17 


.4731764 


.6750231 


.5133732 


.7104332 


67 


.0523852 


.7192085 


.0722381 


.8587663 


18 


.452S004 


.6559068 


.4936281 


.6933999 


68 


.0501294 


.7000922 


.0694597 


.8417329 


19 


.4333018 


.6367905 


.4746424 


.6763666 


69 


.0479707 


.6809760 


.0667882 


.8246996 


20 


.4146429 


.6176742 


.4563869 


.6593332 


70 


.0459050 


.6618597 


.0642194 


.8076662 


21 


.3967874 


.5985579 


.4388336 


.6422999 


71 


.0439282 


.6427434 


.0617494 


.7906329 


22 


.3797009 


.5794416 


.4219554 


.6252665 


72 


.0420366 


.6236271 


.0593744 


.7735996 


28 


.3633501 


.5603253 


.4057263 


.6082332 


73 


.0402264 


.6045108 


.0570908 


.7565662 


24 


.3477035 


.5412090 


.3901215 


.5911999 


74 


.0384941 


.5853945 


.0548950 


.7395329 


23 


.3327306 


.5220927 


.3751168 


.5741665 


75 


.0368365 


.5662782 


.0527837 


.7224996 


2G 


.3184025 


.5029764 


.3606892 


.5571332 


76 


.0352502 


.5471619 


.0507535 


.7054662 


27 


.3046914 


.4838602 


.3468166 


.5400998 


77 


.0337323 


.5280456 


.0488015 


.6884329 


28 


.2915707 


.4647439 


.3334775 


.5230665 


78 


.0322797 


.5089293 


.0469245 


.6713995 


29 


.2790150 


.4456276 


.3206514 


.5060332 


79 


.0308897 


.4898131 


.0451197 


.6543662 


30 


.2670000 


.4265113 


.3083187 


.4889998 


80 


.0295595 


.470696S 


.0433843 


.6373329 


31 


.2555024 


.4073950 


.2964603 


.4719665 


81 


.0282866 


.4515805 


.0417157 


.62C2995 


82 


.2444999 


.3882787 


.2850519 


.4549331 


82 


.0270685 


.4324642 


.0401112 


.6032662 


S3 


.2339712 


.3691624 


.2740942 


.4378998 


83 


.0259029 


.4133479 


.0385685 


.5862328 


C4 


.2238959 


.3500461 


.2635521 


.420S665 


84 


.0247874 


.3942316 


.0370851 


.5691995 


33 


.2142544 


.3309298 


.2534155 


.4038331 


85 


.0237200 


.3751153 


.0356587 


.5521662 


36 


.2050282 


.3118135 


.2436687 


.3867998 


86 


.0226986 


.3559990 


.0342873 


.5351328 


37 


.1961992 


.2926973 


.2342968 


.3697664 


87 


.0217211 


.3368827 


.0329685 


.5180995 


38 


.1877504 


.2735810 


.2252854 


.3527331 


88 


.0207858 


.3177664 


.0317005 


.5010661 


30 


.1796655 


.2544647 


.2166206 


.3356998 


89 


.0198907 


.2986502 


.0304812 


.4840328 


40 


.1719287 


.2353484 


.2082890 


.3186664 


90 


.0190342 


.2795339 


.0293089 


.4669995 


41 


.1645251 


.2162321 


.2002779 


.3016331 


91 


.0182145 


.2604176 


.0281816 


.4499661 


42 


.1574403 


.1971158 


.1925749 


.2845997 


92 


.0174302 


.2413013 


.0270977 


.4329328 


43 


.1506605 


.1779995 


.1851682 


.2675664 


93 


.0166796 


.2221850 


.0260555 


.4158994 


44 


.1441728 


.1588832 


.1780463 


.2505331 


94 


.0159613 


.2030687 


.0250534 


.3988661 


45 


.1379644 


.1397669 


.1711984 


.2334997 


95 


.0152740 


.1839524 


.0240898 


.3818328 


46 


.1320233 


.1206506 


.1646139 


.2164664 


96 


.0146163 


.1648361 


.0231632 


.3648000 


47 


.1263381 


.1015343 


.1582826 


.1994331 


97 


.0139868 


.1457198 


.0222723 


.3477667 


48 


.1208977 


.0824181 


.1521948 


.1823997 


98 


.0133845 


.1266035 


.0214157 


.3307333 


40 


.1156916 


.0633018 


.1463411 


.1653664 


99 


.0128082 


.1074872 


.0205920 


.3137000 


50 


.1107096 


.0441855 


.1407126 


.1483330 


100 


.0122566 


.0883710 


.0198000 


.2966667 



178 



NOTES ON LIFE INSURANCE. 



Actuaries' Table of Mortality. 









Percentage 








Percent'ge 


Age. 


Living. 


Deaths. 


of deaths 

to number 

living. 


Age. 


Living. 


Deaths. 


of deaths 

to number 

living. 


10 


100000 


676 


.00676 


55 


63469 


1375 


.02186 


11 


99324 


674 


.00679 


56 


62094 


1436 


.02313 


12 


98650 


672 


.00681 


57 


60658 


1497 


.0246S 


13 


97978 


671 


.00685 


58 


59161 


1561 


.02639 


14 


97307 


671 


.00690 


59 


57600 


1627 


.02825 


15 


96636 


671 


.00694 


60 


55973 


1698 


1 .03034 


16 


95965 


672 


.00700 


61 


54275 


1770 


.03261 


17 


95293 


673 


.00706 


62 


52505 


1844 


.03512 


18 


94620 


675 


.00713 


63 


50661 


1917 


.03784 


19 


93945 


677 


.00721 


64 


48744 


1990 


.04083 


20 


93268 


680 


.00729 


65 


46754 


2061 


.04408 


21 


92588 


683 


.00738 


66 


44693 


2128 - 


.04761 


22 


91905 


686 


.00746 


67 


42565 


2191 


.05147 


23 


91219 


690 


.00756 


68 


40374 


2246 


.05563 


24 


90529 


694 


.00767 


69 


38128 


2291 


.06009 


25 


89835 


698 


.00777 


70 


35837 


2327 


.06493 


26 


89137 


703 


00789 


71 


33510 


2351 


.07016 


27 


88434 


708 


.00801 


72 


31159 


2362 


.07580 


28 


87726 


714 


.00814 


73 


28797 


2358 


.08188 


29 


87012 


720 


.00827 


74 


26439 


2339 


.08847 


30 


86292 


727 


.0OS42 


75 


24100 


2303 


.09556 


31 


85565 


734 


.00858 


76 


21797 


2249 


.10318 


32 


84831 


742 


.00875 


77 


19548 


2179 


.11147 


33 


84089 


750 


.00892 


78 


17369 


2092 


.12044 


34 


83339 


758 


.00909 


79 


15277 


1987 


.13006 


35 


82581 


767 


.00929 


80 


13290 


1866 


.14041 


36 


81814 


776 


.00948 


81 


11424 


1730 


.15144 


37 


81038 


785 


.00969 


82 


9694 


1582 


.16319 


38 


80253 


795 


.00991 


83 


8112 


1427 


.17591 


19 


79458 


805 


.01013 


84 


6685 


1268 


.18968 


40 


78653 


815 


.01036 


85 


5417 


1111 


.20509 


41 


77838 


826 


.01061 


86 


4306 


958 


.22248 


42 


77012 


839 


.01089 


87 


3348 


811 


.24223 


43 


76173 


857 


.01125 


88 


2537 


673 


.26527 


44 


75316 


881 


.01170 


89 


1864 


545 


.29233 


45 


74435 


909 


.01221 


90 


1319 


427 


.32373 


46 


73526 


944 


.01284 


91 


892 


322 


.36099 


47 


72582 


981 


.01352 


92 


570 


231 


.40526 


48 


71601 


1021 


.01426 


93 


339 


155 


.45723 


49 


70580 


1063 


.01506 


94 


184 


95 


.51630 


50 


69517 


1108 


.01594 


95 


89 


52 


.58427 


51 


68409 


1156 


.01690 


96 


37 


24 


.64865 


52 


67253 


1207 


.01795 


97 


13 


9 


.69231 


53 


66046 


1261 


.01909 . 


98 


4 


3 


.75000 


54 


64785 


1316 


.02031 


99 

1 


1 


1 


1.00000 



NOTES OX LIFE INSUKANCE. 



179 



American Mcperience Table of Mortality. 



Age. 


Number 


Number of 


Percentage 

of deaths to 


Age. 


Number 


Number of 


Percent- 
age of 


living. 


deaths. 


living. 


living 


deaths. 


deaths to 














living. 


10 


100000 


749 


.00749 


53 


66797 


1091 


.01633 


11 


99251 


746 


.00752 


54 


65706 


1143 


.01735 


12 


98505 


743 


.00754 


55 


64563 


1199 


.01857 


13 


97762 


740 


.00757 










14 


97022 


737 


.00760 


. 56 


63364 


1260 


.01988 


15 


96285 


735 


.00763 


57 


62104 


1325 


.02134 










58 


60779 


1394 


.02293 


16 


95550 


732 


.00766 


59 


59385 


1468 


.02472 


17 


94818 


729 


.00769 


60 


57917 


1546 


.02669 


18 


94089 


727 


.00773 










19 


93362 


725 


.00777 


61 


56371 


1628 


.02888 


20 


92637 


723 


.00780 


62 


54743 


1713 


.03129 










63 


53030 


1800 


.03394 


21 


91914 


722 


.00786 


64 


512S0 


1889 


.03687 


22 


91192 


721 


.00791 


65 


49341 


1980 


.04013 


23 


90471 


720 


.00796 










24 


89751 


719 


.00801 


66 


47361 


2070 


.04371 


25 


89032 


718 


.00806 


67 


45291 


2158 


.04765 










68 


43133 


22-13 


.05200 


2S 


8S314 


718 


.00813 


69 


40890 


2321 


.05676 


27 


87596 


718 


.00820 


70 


38569 


2391 


.06199 


28 


86878 


718 


.00826 










29 


£6160 


719 


.00834 


71 


36178 


2448 


.06767 


30 


85441 


720 


.00843 


72 


33730 


2487 


.07373 










73 


31243 


2505 


.08018 


31 


84721 


721 


.00851 


74 


28738 


2501 


.08703 


32 


84000 


723 


.00861 


75 


26237 


2476 


.09437 


33 


83277 


726 


.00892 










34 


82551 


729 


.00883 


76 


23761 


2431 


.10231 


35 


81822 


732 


.00895 


77 


21330 


2369 


.11102 










78 


18961 


2291 


.12083 


36 


81090 


737 


.00909 


79 


16670 


2196 


.13173 


37 


80353 


742 


,00923 


80 


14474 


2091 


.14447 


38 


79611 


749 


.00941 










39 


78862 


756 


.00959 


81 


12383 


1964 


.15860 


40 


78106 


765 


.00979 


82 


104*19 


1816 


.17429 










83 


8603 


1648 


.19156 


41 


77341 


774 


.01001 


84 


6955 


1470 


.21136 


42 


76567 


785 


.01025 


85 


5485 


1292 


.23555 


43 


75782 


797 


.01052 










44 


74985 


812 


.01083 


88 


4193 


1114 


.26568 


45 


74173 


828 


.01116 


87 


3079 


933 


.30302 










88 


2146 


744 


.34669 


46 


73345 


848 


.01156 


89 


1402 


555 


.39515 


47 


72497 


870 


.01200 


90 


847 


385 


.45455 


48 


71627 


896 


.01251 










49 


70731 


927 


.01316 


91 


462 


246 


.53247 


£0 


69804 


962 


.01378 


92 


216 


137 


.63426 










93 


79 


58 


.73418 


51 


6S842 


1001 


.01454 


94 


21 


18 


.85714 


52 


67841 


1044 


.01539 


95 


3 


3 


1.00000 



180 



NOTES ON LIFE INSURANCE. 



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S-«2^SSSSSS^^S5^^^^g5f?8coc??S5?i?^5geo§^^^^§^^?§S 



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181 



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182 



NOTES ON LIFE INSURANCE. 






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183 



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184 



NOTES ON LIFE INSURANCE. 






A, COLUMNS. 

Value at Age x of a Life Series of Annual Payments of $1 each 
— the first immediate. 





American Ex- 








American Ex- 






Age. 


perience 


Age. 


Actuaries' 


Age. 


perience 


Age. 


Actuaries' 




4>£ per cent. 




4 per cent. 




4J4 per cent. 




4 per cent. 


10 


18.855 2865 


10 


20.4536 


55 


11.821 8465 


55 


11.9779 


11 


18.799 5832 


11 


20.3694 


56 


11.522 8199 


56 


11.6698 


12 


18.741 4307 


12 


20.2818 


57 


11.219 4467 


57 


11.3593 


13 


18.680 6991 


13 


20.1907 


58 


10.912 1342 


58 


11.0463 


14 


18.617 2521 


14 


20.0959 


59 


10.601 3275 


59 


10.7311 


15 


18.550 9454 


15 


19.9976 


60 


10.287 6997 


60 


10.4147 


16 


18.481 8206 


16 


19.8957 


61 


9.971 8279 


61 


10.0977 


17 


18.409 5363 


17 


19.7901 


62 


9.654 3796 


62 


9.7805 


18 


18.333 9242 


18 


19.6807 


63 


9.335 9646 


63 


9.4641 


19 


18.255 0022 


19 


19.5675 


64 


9.017 1527 


64 


9.1489 


20 


18.172 5961 


20 


19.4504 


65 


8.698 6699 


65 


8.&356 


21 


18.086 5220 


21 


19.3293 


66 


8.381 4484 


66 


8.524S 


22 


17.996 '!833 


22 


19.2042 


67 


8.066 1600 


67 


8.2170 


23 


17.903 1882 


23 


19.0747 


68 


7.753 5753 


68 


7.9130 


24 


17.805 5344 


24 


18.9410 


69 


7.444 6209 


69 


7.6130 


25 


17.703 6080 


25 


18.8027 


70 


7.139 9044 


70 


7.3172 


26 


17.597 1831 


26 


18.6598 


71 


6.840 2460 


71 


7.0261 


27 


17.486 2208 


27 


18.5122 


72 


6.545 9947 


72 


6.7400 


28 


17.370 4817 


28 


18.3597 


73 


6.256 9020 


73 


6.4593 


29 


17.249 7130 


29 


18.2022 


74 


5.972 3102 


74 


6.1840 


30 


17.123 8475 


30 


18.0397 


75 


5.691 3706 


75 


5.9146 


31 


16.992 6152 


31 


17.8718 


76 


5.413 3424 


76 


5.6512 


32 


16.855 7300 


32 


17.6985 


77 


5.137 5702 


77 


5.3938 


33 


16.713 0898 


33 


17.5196 


78 


4.863 9745 


78 


5.1428 


3i 


16.564 5871 


34 


17.3350 


79 


4.592 7856 


79 


4.8986 


35 


16.409 9079 


35 


17.1443 


80 


4.324 0890 


80 


4.6607 


36 


16.248 7188 


36 


16.9476 


81 


4.060 2393 


81 


4.4290 


37 


16.081 0666 


37 


16.7443 


82 


3.800 7694 


82 


4.2026 


38 


15.906 6002 


38 


16.5342 


83 


3.544 6260 


83 


3.9802 


39 


15.725 3451 


39 


16.3172 


84 


3.289 2139 


84 


3.7611 


40 


15.536 9284 


40 


16.0929 


85 


3.033 3545 


85 


3.5436 


41 


15.341 3492 


41 


15.8610 


86 


2.779 5927 


86 


3.3279 


42 


15.138 2074 


42 


15.6212 


87 


2.532 5153 


87 


3.1138 


43 


14.927 4700 


43 


15.3736 


88 


«.297 7410 


88 


2.9012 


44 


14.708 8998 


44 


15.1186 


89 


2.075 8025 


89 


2.6911 


45 


14.482 6303 


45 


14 8571 


90 • 


1.860 8589 


90 


2.4S54 


46 


14 248 4049 


46 


14.5896 


91 


1.649 2622 


91 


2.2843 


47 


14.006 5237 


47 


14.3170 


92 


1.451 1912 


92 


2.0902 


48 


13.756 9070 


48 


14.0394 


93 


1.289 1504 


93 


1.9065 


49 


13.499 8407 


49 


13.7572 


94 
95 


1.136 7054 
1.000 0000 


94 


1.7369 


50 


13.235 8018 


50 


13.4703 






95 


1.5843 


51 


12.965 0906 


51 


13.1792 






96 


1.4618 


52 


12.6S8 0262 


52 


12.8841 






97 


1.3670 


53 


12.404 8683 


53 


12.5853 






98 


1.2404 


54 


12.115 9785 


54 | 


1/.2S32 






99 


1.0000 



NOTES ON LIFE INSURANCE. 



1S5 



Actuaries' four per cent 



For Use in Accumulation Formulas. 



Age. 


tfe. 


Cx. 


ft* 


Age. 


Vx. 


Cx. 


k x . 


10 


1.04708 


.006500 


.006806 


55 


1.06303 


.020831 


.022144 


11 


1.04710 


.006525 


.006832 


56 


1.06462 


.022237 


.023674 


12 


1.04713 


.006550 


.006859 


57 


1.06632 


.023730 


.025304 


13 


1.04717 


.006585 


.006896 


58 


1.06819 


.025371 


.027101 


14 


1.04722 


.006630 


.006944 


59 


1.07023 


.027160 


.029068 


15 


1.04727 


.006677 


.006992 


60 


1.07254 


.029169 


.031285 


16 


1.04733 


.006733 


.007052 


61 


1.07506 


.031357 


.033711 


17 


1 .04740 


.006791 


.007113 


62 


1.07786 


.033770 


.036399 


18 


1.04747 


.006860 


.007185 


63 


1.08090 


.036384 


.039328 


19 


1.04755 


.006929 


.007259 


64 


1.08427 


.039255 


.042563 


20 


1.04764 


.007010 


.007344 


65 


1.0S796 


.042386 


.046115 


21 


1.04773 


.007093 


.007432 


66 


1.09199 


.045782 


.049994 


22 


1.04782 


.007177 


.007520 


67 


1.09644 


.049494 


.054268 


33 


1.04793 


.007273 


.007622 


68 


1.10126 


.053490 


.058907 


24 


1.04803 


.007371 


.007725 


69 


1.10648 


.057776 


.063928 


25 


1.04814 


.007471 


.007830 


70 


1.11222 


.062436 


.069442 


26 


1.04827 


.007583 


.007949 


71 


1.11847 


.067460 


.075452 


27 


1.04839 


.007698 


.008071 


72 


1.12530 


.072889 


.082022 


28 


1.04854 


.007826 


.008206 


73 


1.13275 


.078734 


.089186 


29 


1.04868 


.007956 


.008344 


74 


1.14093 


.085065 


.097054 


30 


1.04884 


.008101 


.008497 


75 


1.14988 


.091885 


.105657 


31 


1.04900 


.00824S 


.008652 


76 


1.15965 


.099211 


.115050 


32 


1.04918 
1.04936 


.008410 


.008824 


77 


1.17047 


.107182 


.125453 


33 


.008576 


.008999 


78 


1.18242 


.115812 


.136938 


34 


1.04955 


.008746 


.009179 


79 


1.19549 


.125062 


.149511 


35 


1.04975 


.008931 


.009375 


80 


1.20987 


.135006 


.163340 


36 


1.04996 


.009122 


.009576 


81 


1.22560 


.145611 


.179492 


37 


1.05017 


.009314 


.009782 


82 


1.24282 


.156917 


.195020 


38 


1.05040 


.009525 


.010005 


83 


1.26200 


.169146 


.213463 


39 


1.05064 


.009741 


.010235 


84 


1.28344 


.182383 


.234078 


40 


1.05089 


.009963 


.010470 


85 


1.30833 


.197207 


.258012 


41 


1.05116 


.010204 


.010726 


86 


1.33759 


.213923 


.2S6141 


42 


1.05146 


.010476 


.011014 


87 


1.37246 


.232917 


.319669 


43 


1.05183 


.010818 


.011379 


88 


1.41549 


.255071 


.361052 


44 


1.05231 


.011247 


.011836 


89 


1.46972 


.281137 


.413192 


45 


1.05286 


.011742 


.012363 


90 


1.53785 


.311279 


.478700 


46 


1.05353 


.012345 


.013006 


91 


1.62751 


.347103 


.564912 


47 


1.05425 


.012996 


.013701 


92 


1.74867 


.389676 


.681416 


48 


1.05504 


.013711 


.014466 


93 


1.91609 


.439641 


.842391 


49 


1.05590 


.014482 


.015291 


94 


2.15011 


.496446 


1.067416 


50 


1.05684 


.015326 


.016197 


95 


2.50162 


.561798 


1.405405 


51 


1.05788 


.016248 


.017189 


96 


2.95999 


.623701 


1.846154 


52 


1.05901 


.017257 


.018275 


97 


3.37999 


.6656S0 


2.250000 


53 


1.06024 


.018359 


.019464 


98 


4.15999 


.721154 


3.000000 


54 


1.06156 


.019532 


.020735 


99 




.961538 





1S6 



NOTES OX LIFE INSURANCE. 



American Experience Four and a half per cent. 

For "Use in Accumulation Formulas. 



d 
to 


Ut. 


% 


Cz. 


b£ 


*, 


< 




< 




< 




10 


1.0528S6 


10 


.00716746 


10 


.0075465 ' 


11 


1.052914 


11 


.00719261 


11 


.0075732 


12 


1.052942 


12 


.00721796 


12 


.0076001 


13 


1.0529 TO 


13 


.00724345 


13 


.0076271 


14 


1.052999 


14 


.00726911 


14 


.0076544 


15 


1.053038 


15 


.00730487 


15 


.0076923 


16 


1.053067 


16 


.00733101 


16 


.0077201 


17 


1.053097 


17 


.097:35733 


17 


.0O774S0 i 


IS 


1.053137 


13 


.00739400 


IS 


.0077869 ' 


19 


1.053173 


19 


.00743107 


19 


.0078262 


23 


1.O53220 


20 


.00746857 


20 


0078660 


21 


1.053274 


21 


.00751691 


21 


.0079174 


22 


1.053328 


22 


.00756593 


22 


.0079694 


23 


1.0533S8 


23 


.00761565 


23 


.0080222 


24 


1.053439 


24 


.00766603 


24 


.0080757 


25 


1.053493 


25 


.03771724 


25 


.0081301 


25 


1.(63568 


26 


.00777996 


25 


.0081967 


27 


1.053633 


27 


.0)7-4375 


27 


.0082645 


23 


1.053708 


23 


.00790358 


28 


.0083:333 


29 


1.053794 


29 


.00798559 


29 


.0084152 


30 


1.053831 


3) 


.00806393 


30 


.0084985 


31 


1.053970 


31 


.03814382 


31 


.0085833 


32 


1.054073 


32 




32 


.0086819 


33 


1.05419'J 


33 


.003:34243 


^ 


.0087946 


34 


1. 054311 


34 


.00345063 


34 


.0089096 


35 


1.054433 


35 


.00356100 


.35 


.0090270 | 


38 


1.054535 


33 


.00869729 


36 


.0091720 : 


37 


1.054740 


37 




37 


.0093203 


33 


1.054925- 


S3 


.00900311 


38 


.0094976 1 


39 


1.055 115 


39 


.00917356 


33 


.0096792 


40 


1.055336 


40 


.00937261 


40 


.009S913 


41 


1.055554 


41 


.00957663 


41 


.010103S ; 


42 


1.055325 


42 


.00981097 


42 


.0103587 


43 


1.056107 


43 


.01006412 


43 


.0106288 


44 


1.056440 


44 


.01036252 


44 


.0109474 


45 


1.036797 


45 


.01068238 


45 


.0112391 


46 


1.057223 


46 


.01106392 


46 


.0116970 


47 


1.057693 


47 


.01148373 


47 


.0121463 


43 


1.058238 


43 


.01197057 


48 


.0126677 


49 


1.053878 


49 


.01254162 


49 


.0132800 


50 


1.059603 


50 


.01318799 


50 


.0139740 


51 


1.060419 


51 


.01391439 


51 


.0147551 


52 


1.061333 


52 


.0147-2624 


. 


.0156294 : 



bo 


Vz. 


6 

3? 


Cz. 




7;z. 


< 




< 




< 




53 


1.062351 


53 


.01562973 


53 


.0166043 


54 


1.063500 


54 


.01664653 


54 


.0177036 


55 


1.064774 


55 


.01777130 


55 


.0189224 


56 


1.066202 


56 


.01902381 


50 


.0202885 


57 


1.067781 


57 


. 02041 K44 


57 


.0218013 


53 


1.069530 


53 


.02194790 


53 


.0234739 


59 


1.071487 


59 


.0236.5555 


59 


.0253456 


60 


1.073660 


60 


.02554390 


60 


.0274234 


61 


1.076077 


61 


.C2763646 


61 


.0-297390 


62 


1.078756 


62 


.02994418 


62 


.0323025 


63 


1.081717 


63 


.03243139 


63 


.0:351357 


64 


1.085007 


64 


.03528510 


64 


.0382846 


65 


1.088688 


65 


.03840086 


65 


.0418042 


C6 


1.092761 


66 


.04182473 


66 


.0457044 


67 


1.097283 


67 


.C4359563 


67 


.050*313 


68 


1.102323 


68 


.04976263 


63 


.0548545 


69 


1.107886 


69 


.05431775 


69 


.0601779 


70 


1.114064 


70 


.05932325 


70 


.0660899 


71 


1.120842 


71 


.06475161 


71 


.0725763 


72 


1.128184 


72 


.07055749 


72 


.0796013 


73 


1.136089 


73 


.07672532 


73 


.0871653 


74 


1.144613 


74 


.08328003 


•74 


.0953234 


75 


1.153894 


75 


.09030674 


75 


.1042044 


76 


1.164100 


76 


.09790479 


76 


.1139709 


77 


1.17556-3 


77 


.10628156 


77 


.1249407 


78 


1.188617 


78 


.11562389 


78 


.1374325 


79 


1.203548 


79 


.12606091 


79 


.1517200 


80 


1.221459 


80 


.13824492 


SO 


.1688603 


81 


1.241984 


SI 


.15177468 


81 


.1585018 


82 


1.265588 


82 


.16679135 


82 


.2110892 


83 


1.292615 


83 


.183:31204 


83 


.2369518 


84 


1.325064 


84 


.20225716 


84 


.2680036 


85 


1.366999 


85 


.22540814 


85 


.3031324 


86 


1.423087 


S6 


.25424009 


86 


.3618953 


87 


1.499327 


-7 


.28997173 


87 


.4347624 


S3 


1.599551 


8 


.33176222 


88 


.5306705 


89 


1.729740 


89 


.37881632 


89 


.6552333 


90 


1.915833 


90 


.43497173 


90 


.8333333 


91 


2.235139 


91 


.50953831 


91 


1.138S839 


92 


2.857215 


92 


.60694666 


92 


1.7341772 


93 


3.931190 


93 


.70256193 


93 


2.7619048 


94 


7.315000 


94 


.82023240 


94 


6.0000000 


95 




95 


.95603780 







NOTES ON L1PE INSURANCE. 



is: 



J)ecimcds of a year from Jute to January 1st. 





Jan. 


Feb. 


Mar. 


April. 


May. 


Jane. 


July. 


Aug. 


Sept. 


Oct. 


Nov. 


Dec. 


1 


1.00000 


91507 


8G83G 


75342 


67123 


58630 


50411 


41918 


88428 


252C5 


16712 


08493 


2 


99726 


91233 


8S5G2 


75068 


G6849 


C835G 


50137 


41644 


88151 


24932 


16438 


08219 


a 


99452 


90959 


83288 


74795 


G657S 


58082 


49868 


41370 


S2S77 


24658 


16164 


07945 


4 


9917S 


90085 


83014 


74521 


. 66301 


57S08 


49589 


41096 


326C3 


24384 


15890 


07671 


5 


98904 


904 11 


82740 


71247 


CCG27 


57534 


49315 


40822 


S2829 


24110 


15616 


07897 


6 


9S630 


90137 


82466 


73973 


C5753 


57260 


49041 


40548 


S2055 


23836 


15342 


07123 


7 


9836G 


89863 


82192 


73699 


65479 


56986 


48767 


40274 


31781 


23562 


15063 


00849 


8 


9S0S2 


895S9 


81918 


73425 


65205 


56712 


48493 


40000 


31507 


23288 


14795 


06575 


9 1 


97808 


89315 


81644 


73151 


64982 


56433 


48219 


39726 


31233 


23014 


14521 


C6301 


10 


97534 


69C41 


81370 


72677 


64658 


561G4 


47945 


39452 


30959 


22740 


14247 


C6C27 


11 


97260 


8S7G7 


G109G 


72603 


64384 


55S90 


47671 


39178 


30G85 


22466 


13973 


05753 


12 


9698G 


88493 


80822 


72329 


64110 


5561G 


47397 


38904 


30411 


22192 


13693 


C5479 


13 


96712 


88219 


80548 


72055 


63S36 


55342 


47123 


3S630 


30137 


21918 


13425 


05205 


14 


96438 


87945 


80274 


71781 


63562 


55068 


46849 


48356 


29863 


21644 


13151 


04932 


15 


961G4 


87G71 


80000 ; 


71507 


63288 


54795 


46575 


38082 


29589 


21370 


12877 


04658 


10 


95890 


87397 


1 
79728 


71233 


63014 


54521 


46301 


378C8 


29315 


21096 


12603 


04384 


17 


95616 


87123 


79452 1 


70959 


62740 


54247 


46027 


37534 


29041 


20822 


12329 


04110 


IS 


£5342 


86849 


79178 \ 


7C685 


62466 


53973 


45753 


37260 


28767 


20548 


12055 


038SG 


19 


95C6S 


86575 


78904 [ 


70411 


62192 


53699 


45479 


3698G 


28493 


20274 


11781 


0S562 


~° 


94795 


8G301 


78GS0 ! 
i 


70137 


61918 


53425 


45205 


36712 


28219 


! 20000 


11507 


03288 


« 


84521 


86C27 


! 
78356 1 


69863 


61644 


53151 


44932 


36438 


27945 


19726 


11233 


C3014 


22 


94247 


85753 


7S082 


69589 


61370 


52877 


44658 


36164 


27671 


19452 


10959 


C2740 


23 


93973 


85479 


77808 


69315 


61096 52603 


443S4 


35890 


27397 


19178 


10C85 


0246C 


24 


93C99 


85205 


77534 


69041 


60822 52329 


44110 


35616 


27123 


18904 


10411 


02192 


25 


93425 


84932 


77260 


68767 


6C548 


52055 


43836 


35342 


26849 


18630 


10137 


01918 


20 


93151 


84658 


76986 


68493 


60274 


51781 


43562 


35068 


2G575 


1835G 


C9863 


01644 


27 


92877 


84384 


76712 


68219 


600C0 


51507 


44288 


34795 


26301 


18082 


C9539 


01370 


28 


92603 


84110 


76438 


67945 


59726 


51233 


43014 


34521 


26027 


17808 


09315 


01096 


29 


92329 


84110 


76164 


67671 


59452 


50959 


42740 


34247 


25753 


17534. 


C9041 


00822 


CO 


92055 




75890 


67397 


59178 


50685 


42466 


33973 


25479 


17260 


C8767 


00548 


31 


917S1 




75616 




53904 


J 42192 


33099 




16986 




00274 




Jen. 


1 
Feb. Mar. 


April. 


May. I June. 


i July. 


Aug. 


Sept. 


1 
Oct. 


Nov. Dec. 



INDEX. 



TAGK 

Accounts of life companies Ill 

Actuaries' table of mortality 15 

Actuary of the Royal Exchange Assurance office 129, 130 

Additional insurance purchased with surplus 113 

Algebraic discussion 66 

Algebraic summary 166 

Age of insured taken to be nearest whole years 122 

Agents should explain the policy they sell 131 

Amalgamations, reinsuring 120 

American experience table of mortality 15 

Amount that will produce $1 in one year 13, 68 

Amount that will produce $1 in n years 14, 68, 70 

Amount that will insure $1 for one year 17, 18, 71 

Amount that will insure $1 for n years 24, 70 

Amount that will insure $1 for life 21, 72 

Amount at risk 50 

Annual premiums, whole life 27, 81, 73 

Annual premiums for n years 32, 73 

Annual statements 141 

Aunuities paid oftener than once a year 100 

Anticipating future profits, treating them as assets 130 

Appendix, algebraic summary 166 

Appendix, extracts from Masseres 147 

Appendix, formulas for deposit or " reserve" 171 

Appendix, quotations from actuaries 160 

Appendix, list of tables in 175 

aP x , meaning of and expression for 73 

aP x> in terms of A x 76 

Assets 143 

A z , meaning of and general expression for 73 

A x columns 184 

Balance-sheet 141 

Barnes, William 7 

Bartlett, William H. C 5 

Ruckner, S. B 7 

C column, how computed 34 

Campaign literature 120 

Certainty of payment is what policy-holders want 117 

Chance that the insured may die in any year 16, 17, 18 

Chance that the insured will be alive at the beginning of any year 28 

Commutation columns, how computed 34, 62 

Commutation tables, appendix 180, 182 

Company may charge too little 116 

Comparison, actual, with table mortality 119 

Conditions expressed in contract 116 

Contents 9 

Contribution plan 110 



190 INDEX. 

Co-operative life insurance 128 

Cost of insurance on amount at risk 51 

Cost of insurance on $1000 for one jeax 18 

Curtate commutation columns \-'\ _■ 

C x columns 15-J. I-i 

t^= v T columns ISo. L^j 

D column. Low computed $4, 35 

Decreasing premiums ftj, 12a 

Deduction from premiums, miscalled " dividend " Ill 

Deposit or " reserve/' general discussion -C -_-- 

Deposit, decreasing premiums : . 

Deposit, disposition of, when renewal premium is not paid 138 

Deposit, general and special formulas for 171 

Deposit, illustrative example 45 

Descriptive list of policies in force 143 

Lied calculation of net single premium S3, Si 

Detailed calculation, annual life payments $1 each 28, 29 

" Discount the number living '" 160 

Dividends to policv-holders Ill, 114 

Doctrine of chances 60, SO, 14S 

D x columns 180, 182 

4r, meaning of 70 

Endowment 40,74126, 127 

Endowment combined with term insurance 40, 74 

Estimates Ill 

Examination of companies 142 

Expenses 109, 110, 111 

Extracts from Masses z - 147 to 159 

Faup,, Dr. title-page 

Forfeiture for non-payment of premium 116 

Formulas for deposit or "reserve," summary oi 171 

Forty per cent dividend men 118 

Future supposed profits are not realized assets 130 

General management of life companies 109 

GLADSTONE 107, 143 

Gkiswoud, H. A 5 

Gross valuation 129, 130 

Grouping of policies unequal in amount 118 

How much must he in deposit 93 

b^ : (l) (2) (3) ■: : . £•:. ^ 

Insolvency of life companies 133 

Insurance for one year only 17, 122 

Insurance for term of vears 34 

Insurance for life * 21, 73, 124 

Insurance payable at age 75, or death, if prior - 126 

" Insurance value," appendix 161 

Interest and dlscoun: . . IS. .' 7 

Joint lives, annuities, Massebeb 117 

Joint lives, insurance upon 60, 80 

" Keen analysis '" of Mr. , appendix 164 

Knowledge of life insurance needed 131 

hc= ~ = c x Xu x 97 

l- s columns :>:>. li'3 



INDEX. 191 

PAGE 

Lapsed policies 131 

Large deposit or " reserve" means Large debt 46, 47 

Legal standard of safety 131 

Life companies should be controlled by stringent laws 110 

Life companies great money lenders 120 

Life insurance business, nature of, GLADSTONE 107 

Life insurance can be made secure 115 

Life insurance companies may break 115 

Life series, annual payments, £1 each 27, 72, 184 

List of tables in appendix 175 

Loading 109 

l x , meaning of 70 

M column, how computed 34 

Management of life companies 100 

Manner of using mortality tables 16 

Masseres on annuities, one or more lives 147 

Medical examiners 118 

Method of calculating net values in life insurance should be understood.. . 117 

Method of computing annuities, Masseres 159 

Murray, David 6 

Mixed companies 109 

Mutual companies 109 

M x columns 180, 182 

N column, how computed 37 

Net annual premiums, whole life 27, 31, 73 

Net annual premiums, n years 74 

Net annual premiums, term insurance 73 

Net annual return premiums 76 

Net annual return premium and endowment. 89 

Net premiums less than required by legal standard. 45, 133 

Net single premium, wdiole life : 21, 71 

Net single premium, term insurance. 24 

Net single return premium 76 

Net valuation 133 

Net value 43 

Notation 70 

Notices of first edition 5 

Number of plans and schemes 122 

Numerical bragging 117 

N* columns 180, 182 

Overpayment in advance ..110 

Parcieux, table of mortality 148 

Paterson, John 5 

Perry, A. L 6 

Pillsbury, Oliver G 

Plans of insurance 122 

Policy account, illustration of Ill , 

Policy, endowment and insurance 40, 74, 126 

Policy, paid for each year 122 

Policy, paid for by single premium 124 

Policy, paid for by annual premiums, whole life 125 

Policy, paid for by n annual premiums 125 

Policy, paid for by decreasing premiums 125 

Policy, paid for by contributions after death 128 

Policy, return premium plan 126 

Policy, tontine plan 128 

Policy-holder not entitled to withdraw deposit 139 

Policy-holder entitled to insurance for deposit 141 



192 DTDEX. 



Premium notes 113 

" Public indebted to tbe keen analysis of Air. " 164 

Publishers' notices of first edition 5 

Quotations from distinguisbed actuary 160, 161 

Quotation from actuary 160 

B column, bow computed , 38 

r, rate of interest 

r', ratio of interest 94 

Reinsuring and amalgamations 120 

Eelation between *P I? A-, and aV , 7 !i 

Beversionarv value 113 

= «tt 94. 

R x columns ' .' '. .15-0. 1-2 

B : : Inmn, bow computed 38 

Sasford JonsrE.. 3 

Sab :-. Edward 123 

Self-insurance, appendix: 161 

5? ; . meaning of and expression for. 71, 72 

sP x , in terms of A z 76 

S:;/:::liry :: - :^i-:i^i T ; 115 

Stephens, Lnrnm 6 

Stewart. A. P 7 

Stock and mutual rates 114 

Stock companies 109 

Solvency 131 

Surplus". 109 

S x columns ISO, 152 

Tables of mortality 14 148 

Tie-tens, of Kiel." 1 - •' 

Tontine life insurance I - -- 

^=r\- • w 

u s columns ISo, 1>6 

= ^- «*« 

1-f-r 



SO 



<ih)' 

Valuation of policies *» 

Variations from table rate of mortality 113 

Variety in plans of life insurance 122 

Winston, F. S 107 

Wright, Eitzer *» • • • 3 



NOTES ON LIFE INSURANCE. 



PART FIRST— THEORETICAL. 
PART SECOND— PRACTICAL. 



WTTH APPENDIX. 



"The rate of premium which must be charged in order to carry out an insurance con- 
tract is the problem which stands at the threshold of Life Assurance." 

Dr. Fark. 



SECOND EDITION 

REVISED, ENLARGED, AND RE-ARRANGED. 



GUST AY US w: SMITH. 



ISSUED BY 

THE NEW-YORK MUTUAL LIFE INSURANCE CO. 

No. 144 Broadway. 

1875. 



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